Here are a gigantic list of all the codes, sorted alphabetically, that were included in the error correction zoo.

- 120-cell code[1] Spherical \((4,600,(7-3\sqrt{5})/4)\) code whose codewords are the vertices of the 120-cell. See [3][2; Table 1][4; Table 3] for realizations of the 600 codewords.
- 24-cell code[1] Spherical \((4,24,1)\) code whose codewords are the vertices of the 24-cell. Codewords form the minimal lattice-shell code of the \(D_4\) lattice.
- 2D bosonization code[5,6] A mapping between a 2D lattice of qubits and a 2D lattice quadratic Hamiltonian of Majorana modes. This family also includes a super-compact fermionic encoding with a qubit-to-fermion ratio of \(1.25\) [6; Table I].
- 2D color code[7,8] a.k.a. Triangular color code.Color code defined on a two-dimensional planar graph. Each face hosts two stabilizer generators, a Pauli-\(X\) and a Pauli-\(Z\) string acting on all the qubits of the face.
- 2D hyperbolic surface code[9–11] Hyperbolic surface codes based on a tessellation of a closed 2D manifold with a hyperbolic geometry (i.e., non-Euclidean geometry, e.g., saddle surfaces when defined on a 2D plane).
- 2D lattice stabilizer code Lattice stabilizer code in two spatial dimensions.
- 2D subsystem color code[12] a.k.a. 2D gauge color code.A subsystem version of the 2D color code.
- 2T-qutrit code[13] Two-mode qutrit code constructed out of superpositions of coherent states whose amplitudes make up the binary tetrahedral group \(2T\).
- 3D bosonization code[14] A mapping that maps a 3D lattice quadratic Hamiltonian of Majorana modes into a lattice of qubits which realize a \(\mathbb{Z}_2\) gauge theory with a particular Gauss law.
- 3D color code[8] Color code defined on a four-valent four-colorable tiling of 3D space. Logical dimension is determined by the genus of the underlying surface (for closed surfaces) and types of boundaries (for open surfaces).
- 3D fermionic surface code[14–17] a.k.a. 3D toric code with emergent fermion, Levin-Wen fermion model, Fermionic-charge bosonic-loop (FcBl) surface code.A non-CSS 3D Kitaev surface code that realizes \(\mathbb{Z}_2\) gauge theory with an emergent fermion, i.e., the fermionic-charge bosonic-loop (FcBl) phase [18]. The model can be defined on a cubic lattice in several ways [19; Eq. (D45-46)]. Realizations on other lattices also exist [20,21].
- 3D lattice stabilizer code Lattice stabilizer code in three spatial dimensions. Qubit codes are conjectured to admit either fracton phases or abelian topological phases that are equivalent to multiple copies of the 3D surface code and/or the 3D fermionic surface code via a local constant-depth Clifford circuit [19].
- 3D subsystem color code[22] a.k.a. 3D gauge color code.A subsystem version of the 3D color code.
- 3D subsystem surface code[23] a.k.a. 3D subsystem toric code.Subsystem generalization of the surface code on a 3D cubic lattice with gauge-group generators of weight at most three.
- 3D surface code[24,25] a.k.a. 3D toric code, 3D cubic code, Bosonic-charge bosonic-loop (BcBl) surface code.A generalization of the Kitaev surface code defined on a 3D lattice.
- 600-cell code[1] Spherical \((4,120,(3-\sqrt{5})/2)\) code whose codewords are the vertices of the 600-cell. See [26; Table 1][4; Table 3] for realizations of the 120 codewords. A realization in terms of quaternion coordinates yields the 120 elements of the binary icosahedral group \(2I\) [27].
- Abelian LP code[28,29] An LP code for Abelian group \(G\). The case of \(G\) being a cyclic group is a GB code (a.k.a. a quasi-cyclic LP code) [29; Sec. III.E]. A particular family with \(G=\mathbb{Z}_{\ell}\) yields codes with constant rate and nearly constant distance.
- Abelian TQD stabilizer code[30–32] Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order. The corresponding anyon theory is defined by an Abelian group and a Type-III group cocycle that can be decomposed as a product of Type-I and Type-II group cocycles; see [33; Sec. IV.A]. Abelian TQDs realize all modular gapped Abelian topological orders [33]. Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [34].
- Abelian quantum-double stabilizer code[35] Modular-qudit stabilizer code whose codewords realize 2D modular gapped Abelian topological order with trivial cocycle. The corresponding anyon theory is defined by an Abelian group. All such codes can be realized by a stack of modular-qudit surface codes because all Abelian groups are Kronecker products of cyclic groups.
- Abelian topological code Code whose codewords realize topological order associated with an Abelian anyon theory. In 2D, this is equivalent to a unitary braided fusion category which is also an Abelian group under fusion [36]. Unless otherwise noted, the phases discussed are bosonic.
- Accumulate-repeat-accumulate (ARA) code[37] A generalization of the RA code in which the outer repetition-code encoding step is augmented with an acumulator acting on a fraction of the incoming bits. In addition, the code may be punctured after the final acumulating step.
- Additive \(q\)-ary code A \(q\)-ary code whose codewords are closed under addition, i.e., for any codewords \(x,y\), \(x+y\) is also a codeword.
- Alamouti code[38] The simplest OSTBC, with \(n=2\) transmitting antennas, \(m=1\) receiving antennas, and \(t=2\) uses.
- Algebraic LDPC code LDPC code whose parity check matrix is constructed explicitly (i.e., non-randomly) from a particular graph [39,40] or an algebraic structure such as a combinatorial design [41–43], balanced incomplete block design [44], a partial geometry [45], or a generalized polygon [46,47]. The extra structure and/or symmetry [48] of these codes can often be used to gain a better understanding of their properties.
- Algebraic-geometry (AG) code[49–51] Binary or \(q\)-ary code or subcode constructed from an algebraic curve of some genus over a finite field via the evaluation construction, the residue construction, or more general constructions that yield nonlinear codes. Linear AG codes from the first two constructions are also called geometric Goppa codes.
- Alternant code[52–55] Given a length-\(n\) GRS code \(C\) over \(GF(q^m)\), an alternant code is the \(GF(q)\)-subfield subcode of the dual of \(C\); see [56; Ch. 12]. Its parity-check matrix is an alternant matrix.
- Amplitude-damping (AD) code[57,58] Block quantum code on either qubits or bosonic modes that is designed to detect and correct qubit or bosonic AD errors, respectively.
- Amplitude-damping CWS code[59,60] Self-complementary CWS code that is designed to detect and correct AD errors.
- Analog stabilizer code a.k.a. Gaussian stabilizer code, Linear stabilizer code, Symplectic stabilizer code.An oscillator-into-oscillator stabilizer code encoding \(k\) logical modes into \(n\) physical modes. An \(((n,k,d))_{\mathbb{R}}\) analog stabilizer code is denoted as \([[n,k,d]]_{\mathbb{R}}\), where \(d\) is the code's distance.
- Analog surface code[61] a.k.a. Continuous-variable (CV) surface code.An analog CSS version of the Kitaev surface code.
- Analog-cluster-state code[62–64] a.k.a. CV-cluster-state code, CV-graph-state code, Bosonic-cluster-state code.A code based on a continuous-variable (CV), or analog, cluster state. Such a state can be used to perform MBQC of logical modes, which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension. The exact analog cluster state is non-normalizable, so approximate constructs have to be considered.
- Annealing-based spherical code[65–67] Code whose codewords are obtained from a simulated annealing or energy-repulsion numerical optimization procedure.
- Anticode[68,69]
- Antipode lattice code[72] Lattice code constructed via the antipode construction.
- Approximate operator-algebra QECC[73,74] Code encoding quantum and/or classical information that approximately corrects against noise affecting operators forming an algebra.
- Approximate quantum error-correcting code (AQECC)[57,73–78] Encodes quantum information so that it is possible to approximately recover that information from noise up to an error bound in recovery.
- Approximate secret-sharing code[77] A family of \( [[n,k,d]]_q \) CSS codes approximately correcting errors on up to \(\lfloor (n-1)/2 \rfloor\) qubits, i.e., with approximate distance approaching the no-cloning bound \(n/2\). Constructed using a non-degenerate CSS code, such as a polynomial quantum code, and a classical authentication scheme. The code can be viewed as an \(t\)-error tolerant secret sharing scheme. Since the code yields a small logical subspace using large registers that contain both classical and quantum information, it is not useful for practical error correction problems, but instead demonstrates the power of approximate quantum error correction.
- Array code a.k.a. RAID code, Disk array code.Matrix code over \(GF(q)\) designed for use in distributed storage, with the first such application being a RAID-type array of hard-drives such that information is protected against erasure of one or more hard drives. Each column of a matrix codeword is typically treated as a single codeword coordinate via subpacketization (defined below) and represents a large data block. Array codes are denoted by \((n,k,m)\), where \(n\) is the number of nodes, \(k\) is the number of nodes needed to recover the data, and \(m\) is the column dimension of each codeword a.k.a. the subpacketization level.
- Array-based LDPC (AB-LDPC) code[79,80] QC-LDPC code constructed deterministically from a disk array code known as a B-code. Its parity-check matrix admits a compact representation [81] and is related to RS codes.
- Asymmetric quantum code[82,83] a.k.a. Noise-biased quantum code.Quantum systems can be roughly characterized by two types of noise, a bit-flip noise that maps canonical basis states into each other, and a phase-flip noise that induces relative phases between superpositions of such basis states. A code cannot protect against both types of noise arbitrarily well, and there is a tradeoff between the two types of protection. An asymmetric quantum code is one that performs much better against one type of noise than the other type. Such codes typically have tunable distances against each noise type and include CSS codes, GKP codes, and QSCs.
- Auxiliary qubit mapping (AQM) code[84,85] A concatenation of the JW transformation code with a qubit stabilizer code.
- Availability code[86,87] A \(t\)-availability parallel-recovery code is a code such any \(t\) coordinates can be recovered in multiple ways. That way, the code accomodates nodes that may be inaccessible during the recovery procedure.
- B-code[88] The first array code, constructed over \(GF(q)\). See [89] for more details.
- BPSK c-q code Coherent-state c-q binary code encoding into two coherent states \(|\pm\alpha\rangle\) for complex \(\alpha\). A shifted version, with codewords \(\{|0\rangle,|\alpha\rangle\}\), is called binary amplitude modulation (BAM), The three-state subcode \(\{|\alpha,\alpha\rangle,|-\alpha,\alpha\rangle,|\alpha,-\alpha\rangle\}\) of two-mode BPSK is called the single-degeneracy code [90].
- Bacon-Shor code[91,92] Subsystem CSS code defined on an \(m_1 \times m_2\) lattice of qubits that generalizes the \([[9,1,3]]\) (subspace) Shor code. It is said to be symmetric when \(m_1=m_2\), and asymmetric otherwise.
- Balanced code[93] An even-length-\(n\) \(q\)-ary code whose nonzero codewords all have a Hamming weight of \(n/2\). A code is \(\epsilon\)-balanced if the relative weight (i.e., weight divided by \(n\)) of all nonzero codewords lies in the interval \([\frac{1-\epsilon}{2},\frac{1+\epsilon}{2}]\). A code is \(\gamma\)-unbiased if the relative weight lies in the interval \((\frac{1}{2}-\frac{1}{n^{\gamma}},\frac{1}{2}+\frac{1}{n^{\gamma}})\).
- Balanced product (BP) code[94] Family of CSS quantum codes based on products of two classical codes which share common symmetries. The balanced product can be understood as taking the usual tensor/hypergraph product and then factoring out the symmetries factored. This reduces the overall number of physical qubits \(n\), while, under certain circumstances, leaving the number of encoded qubits \(k\) and the code distance \(d\) invariant. This leads to a more favourable encoding rate \(k/n\) and normalized distance \(d/n\) compared to the tensor/hypergraph product.
- Ball color code[95] A color code defined on a \(D\)-dimensional colex. This family includes hypercube color codes (color codes defined on balls constructed from hyperoctahedra) and 3D ball color codes (color codes defined on duals of certain Archimedean solids).
- Ball-Verstraete-Cirac (BVC) code[96,97] a.k.a. Verstraete-Cirac code, Auxiliary fermion code.A 2D fermion-into-qubit encoding that builds upon the JW transformation encoding by eliminating the weight-\(O(n)\) \(X\)-type string at the expense introducing additional qubits. See [6; Sec. IV.B] for details.
- Barnes-Wall (BW) lattice code[98,99] Member of a family of \(2^{m+1}\)-dimensional lattices, denoted as BW\(_{2^{m+1}}\), that are the densest lattices known. Members include the integer square lattice \(\mathbb{Z}^2\), \(D_4\), the Gosset \(E_8\) lattice, and the \(\Lambda_{16}\) lattice, corresponding to \(m\in\{0,1,2,3\}\), respectively.
- Batch code[100] Binary code designed for minimizing the total amount of storage and the worst-case maximal load on any devices in a distributed system.
- Ben-Sasson-Goldreich-Harsha-Sudan-Vadhan (BGHSV) code[101] Locally testable \([n,k,d]\) code with \(n = k^{1+\epsilon}\) and query complexity of order \(O(1/\epsilon)\) for any \(\epsilon > 0\).
- Ben-Sasson-Sudan code[102] Locally testable \([n,k/2,d]_{2^m}\) code with \(k\) a power of two, \(n = k \log^{c} k\), and query complexity \(\log^{c}k\) for some universal constant \(c\).
- Ben-Sasson-Sudan-Vadhan-Wigderson (BSVW) code[103] Locally testable \([n,k,d]\) code with \(n = k \cdot 2^{\tilde{O}(\sqrt{\log k})}\) and asymptotically constant query complexity, where \(\tilde{O}(f)=O(f\cdot (\log f)^c)\) for some fixed constant \(c\).
- Berlekamp code[104; Ch. 9] A linear \(p\)-ary code that has Lee distance 5 and whose construction resembles that of RS codes. It is obtained by first constructing an RS-like parity-check matrix out of a certain field extension of \(GF(p)\) and then taking the subfield subcode of the corresponding code; see [105; Ch. 10.6].
- Best \((10,40,4)\) code[106,107] Binary nonlinear \((10,40,4)\) code that is unique [108]. Under Construction A, this code yields \(P_{10c}\), a non-lattice sphere packing that is the densest known in 10 dimensions [109][110; pg. 140].
- Bicycle code[111] A CSS code whose stabilizer generator matrix blocks are \(H_{X}=H_{Z}=(A|A^T)\), where \(A\) is a circulant matrix. The fact that \(A\) commutes with its transpose ensures that the CSS condition is satisfied. Bicycle codes are the first QLDPC codes.
- Binary BCH code[112–114] Cyclic binary code of odd length \(n\) whose zeroes are consecutive powers of a primitive \(n\)th root of unity \(\alpha\) (see Cyclic-to-polynomial correspondence). More precisely, the generator polynomial of a BCH code of designed distance \(\delta\geq 1\) is the lowest-degree monic polynomial with zeroes \(\{\alpha^b,\alpha^{b+1},\cdots,\alpha^{b+\delta-2}\}\) for some \(b\geq 0\). BCH codes are called narrow-sense when \(b=1\), and are called primitive when \(n=2^r-1\) for some \(r\geq 2\).
- Binary PSK (BPSK) code[115] a.k.a. Binary antipodal modulation, Phase-reversal keying (PRK), Antipodal signaling.Encodes one bit of information into a constellation of antipodal points \(\pm\alpha\) for complex \(\alpha\). These points are typically associated with two phases of an electromagnetic signal.
- Binary antipodal code a.k.a. Binary signal constellation.
- Binary balanced spherical code An \((n-1,K,\frac{nd}{nw-w^2})\) spherical code obtained from a constant-weight-\(w\) binary \((n,K,d)\) code via a component-wise binary balanced mapping (also known as the CW\(_2\) construction), \begin{align} \begin{split} 0&\to\sqrt{\frac{w}{n\left(n-w\right)}}\\1&\to -\sqrt{\frac{n-w}{nw}}~. \end{split} \tag*{(1)}\end{align} This construction can be extended to the general balanced binary construction CW\(_q\) for spherical code alphabets of size \(q\) [116; Sec. 6.6].
- Binary code Encodes \(K\) states (codewords) in \(n\) binary coordinates and has distance \(d\). Usually denoted as \((n,K,d)\). The distance is the minimum Hamming distance between a pair of distinct codewords.
- Binary dihedral PI code[117] Multi-qubit code designed to realize gates from the binary dihedral group transversally. Can also be interpreted as a single-spin code. The codespace projection is a projection onto an irrep of the binary dihedral group \( \mathsf{BD}_{2N} = \langle\omega I, X, P\rangle \) of order \(8N\), where \( \omega \) is a \( 2N \)th root of unity, and \( P = \text{diag} ( 1, \omega^2) \).
- Binary duadic code[118] Member of a pair of cyclic linear binary codes that satisfy certain relations, depending on whether the pair is even-like or odd-like duadic. Duadic codes exist for lengths \(n\) that are products of powers of primes, with each prime being \(\pm 1\) modulo \(8\) [119].
- Binary group-orbit code[120,121] Bianry legnth-\(n\) whose codewords correspond to points in an orbit of some initial vector under a generating group \(G\), which is a subgroup of the group of bit-string permutations and translations, i.e., the automorphism group of binary codes under the Hamming distance.
- Binary linear LTC A binary linear code \(C\) of length \(n\) that is a \((u,R)\)-LTC with query complexity \(u\) and soundness \(R>0\).
- Binary quadratic-residue (QR) code Member of a quadruple of cyclic binary codes of prime length \(n=8m\pm 1\) for \(m\geq 1\) constructed using quadratic residues and nonresidues of \(n\).
- Binary-ternary mixed code[122] Encodes \(K\) states (codewords) in a string of \(n_1+n_2\) coordinates, with the first \(n_1\) coordinates being binary, and the last \(n_2\) coordinates being ternary.
- Binomial code[123] Bosonic rotation codes designed to approximately protect against errors consisting of powers of raising and lowering operators up to some maximum power. Binomial codes can be thought of as spin-coherent states embedded into an oscillator [124].
- Biorthogonal spherical code a.k.a. Cross polytope code, Hyperoctahedron code, Orthoplex code, Co-cube code.Spherical \((n,2n,2)\) code whose codewords are all permutations of the \(n\)-dimensional vectors \((0,0,\cdots,0,\pm1)\), up to normalization. The code makes up the vertices of an \(n\)-orthoplex (a.k.a. hyperoctahedron or cross polytope).
- Bivariate bicycle (BB) code[125] One of several Abelian 2BGA codes which admit time-optimal syndrome measurement circuits that can be implemented in a two-layer architecture, a generalization of the square-lattice architecture optimal for the surface codes.
- Block LDPC (B-LDPC) code[126] Member of a particular class of irregular QC-LDPC codes with efficient encoders.
- Block code A code intended to encode a piece, or block, of a data stream on a block of \(n\) symbols, with each symbol taking values from a fixed alphabet \(\Sigma\).
- Block quantum code A code constructed in a multi-partite quantum system, i.e., a physical space consisting of a tensor product of \(n > 1\) identical factors called subsystems, parties, or bodies. The subsystems include qubits, modular qudits, Galois qudits, oscillators, or more general groups. For finite dimensional codes, the dimension of the underlying subsystem is denoted by \(q\) and is sometimes called the local dimension.
- Body-centered cubic (bcc) lattice code Three-dimensional lattice consisting of all points \((x,y,z)\) whose integer components are either all even or all odd.
- Bose–Chaudhuri–Hocquenghem (BCH) code[127] Cyclic \(q\)-ary code, with \(n\) and \(q\) relatively prime, whose zeroes are consecutive powers of a primitive \(n\)th root of unity \(\alpha\). More precisely, the generator polynomial of a BCH code of designed distance \(\delta\geq 1\) is the lowest-degree monic polynomial with zeroes \(\{\alpha^b,\alpha^{b+1},\cdots,\alpha^{b+\delta-2}\}\) for some \(b\geq 0\). BCH codes are called narrow-sense when \(b=1\), and are called primitive when \(n=q^r-1\) for some \(r\geq 2\). More general BCH codes can be defined for zeroes are powers of the form \(\{b,b+s,b+2s,\cdots,b+(\delta-2)s\}\) where gcd\((s,n)=1\).
- Bosonic \(q\)-ary expansion[128] A one-to-one mapping between basis states on \(n\) prime-dimensional qudits (of dimension \(q=p\)) and the subspace of the first \(p^n\) single-mode Fock states. While this mapping offers a way to map qudits into a single mode, noise models for the two code families induce different notions of locality and thus qualitatively different physical interpretations [129].
- Bosonic c-q code Bosonic code designed for transmission of classical information through non-classical channels.
- Bosonic code a.k.a. Continuous-variable (CV) quantum code, Oscillator code.Encodes logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space that contains at least one oscillator (a.k.a. bosonic mode or qumode).
- Bosonic rotation code[130] a.k.a. Rotationally symmetric bosonic (RSB) code.A single-mode Fock-state bosonic code whose codespace is preserved by a phase-space rotation by a multiple of \(2\pi/N\) for some \(N\). The rotation symmetry ensures that encoded states have support only on every \(N^{\textrm{th}}\) Fock state. For example, single-mode Fock-state codes for \(N=2\) encoding a qubit admit basis states that are, respectively, supported on Fock state sets \(\{|0\rangle,|4\rangle,|8\rangle,\cdots\}\) and \(\{|2\rangle,|6\rangle,|10\rangle,\cdots\}\).
- Bosonic stabilizer code[131,132] a.k.a. CV stabilizer code.Bosonic code whose codespace is defined as the common \(+1\) eigenspace of a group of mutually commuting displacement operators. Displacements form the stabilizers of the code, and have continuous eigenvalues, in contrast with the discrete set of eigenvalues of qubit stabilizers. As a result, exact codewords are non-normalizable, so approximate constructions have to be considered.
- Bosonization code[133–135] A mapping that maps a \(D\)-dimensional lattice quadratic Hamiltonian of Majorana modes into a lattice of qubits. The resulting qubit code can realize various topological phases, depending on the initial Majorana-mode Hamiltonian and its symmetries.
- Bounded-energy code a.k.a. Spherical cluster.Code whose codewords are points lie on or inside a real or complex sphere whose radius squared is called the energy.
- Branching MERA code[136–138] Qubit stabilizer code whose encoding circuit corresponds to a branching MERA tensor network [139].
- Bravyi-Bacon-Shor (BBS) code[140] a.k.a. Generalized Bacon-Shor code.An \([[n,k,d]]\) CSS subsystem stabilizer code generalizing Bacon-Shor codes to a larger set of qubit geometries. Defined through a binary matrix \(A\) such that physical qubits live on sites \((i,j)\) whenever \(A_{i,j}=1\). The gauge group is generated by 2-qubit operators, including \(XX\) interations between any two qubits sharing a column in \(A\), and \(ZZ\) interations between two qubits sharing a row. The code parameters are: \(n=\sum_{i,j}A_{i,j}\), \(k=\text{rank}(A)\), and the distance is the minimum weight of any row or column.
- Bravyi-Kitaev superfast (BKSF) code[141] a.k.a. Loop-stabilized fermion simulation (LSFS) code.An single error-detecting fermion-into-qubit encoding defined on 2D qubit lattice whose stabilizers are associated with loops in the lattice. The code can be generalized to a single error-correcting code (i.e., with distance three) on graphs of degree \(\geq 6\) [142].
- Bravyi-Kitaev transformation (BKT) code[141] A fermion-into-qubit encoding that maps Majorana operators into Pauli strings of weight \(\lceil \log (n+1) \rceil\). The code can be reformulated in terms of Fenwick trees [143], and the Pauli-string weight can be further optimized to yield the segmented Bravyi-Kitaev (SBK) transformation code [144].
- Brown-Fawzi random Clifford-circuit code[145] An \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth \(O(\log^3 n)\).
- CSS-T code[146] A CSS code for which a physical transversal \(T\) gate is either the identity (up to a global phase) or a logical gate. CSS-T codes are constructed from a pair of linear binary codes via the CSS construction, with the pair satisfying certain conditions [147].
- Calderbank-Shor-Steane (CSS) stabilizer code A stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type operators. The two sets of stabilizer generators can often, but not always, be related to parts of a chain complex over the appropriate ring or field.
- Camara-Ollivier-Tillich code[148] A Hermitian qubit QLDPC code whose stabilizer generator matrix is constructed using two nested subgroups of \(GF(4)^n\).
- Cameron-Goethals-Seidel (CGS) isotropic subspace code[149] Member of a \((q(q^2-q+1),(q+1)(q^3+1),2-2/q^2)\) family of spherical codes for any prime-power \(q\). Constructed from generalized quadrangles, which in this case correspond to sets of totally isotropic points and lines in the projective space \(PG_{5}(q)\) [116; Ex. 9.4.5]. There exist multiple distinct spherical codes using this construction for \(q>3\) [150].
- Capped color code (CCC)[151] A non-geometrically local subsystem color code consisting of two layers of 2D color code stacked together and topped (or capped) by a single qubit. Gauge fixing yields two types of codes, capped color codes in H or T form. Layers of 2D color codes can also be stacked together in a recursive construction, yielding recursive capped color codes (RCCCs).
- Cartier code[152] A generalization of the Goppa codes to codes defined from curves of non-zero genus. Each code is a subcode of a certain residue AG code and is constructed using the Cartier operator.
- Cat code[153] a.k.a. Superposition of coherent states (SCS).Rotation-symmetric bosonic Fock-state code encoding a \(q\)-dimensional qudit into one oscillator which utilizes a constellation of \(q(S+1)\) coherent states distributed equidistantly around a circle in phase space of radius \(\alpha\).
- Category-based quantum code Encodes a finite-dimensional logical Hilbert space into a physical Hilbert space associated with a finite category. Codes on modular fusion categories are often associated with a particular topological quantum field theory (TQFT), as the data of such theories is described by such categories.
- Chamon model code[154,155] a.k.a. Chamon-Bravyi-Leemhuis-Terhal (CBLT) code.A foliated type-I fracton non-CSS code defined on a cubic lattice using one weight-eight stabilizer generator acting on the eight vertices of each cube in the lattice [19; Eq. (D38)].
- Chebyshev code[156] Single-mode bosonic Fock-state code that can be used for error-corrected sensing of a signal Hamiltonian \({\hat n}^s\), where \({\hat n}\) is the occupation number operator. Codewords for the \(s\)th-order Chebyshev code are \begin{align} \begin{split} \ket{\overline 0} &=\sum_{k \text{~even}}^{[0,s]} \tilde{c}_k \Ket{\left\lfloor M\sin^2\left( k\pi/{2s}\right) \right\rfloor},\\ \ket{\overline 1} &= \sum_{k \text{~odd}}^{[0,s]} \tilde{c}_k \Ket{\left\lfloor M\sin^2 \left(k\pi/{2s}\right) \right\rfloor}, \end{split} \tag*{(2)}\end{align} where \(\tilde{c}_k>0\) can be obtained by solving a system of order \(O(s^2)\) linear equations, and where \(\lfloor x \rfloor\) is the floor function. The code approaches optimality for sensing the signal Hamiltonian as \(M\) increases.
- Checkerboard model code[157] A foliated type-I fracton code defined on a cubic lattice that admits weight-eight \(X\)- and \(Z\)-type stabilizer generators on the eight vertices of each cube in the lattice.
- Chen-Hsin invertible-order code[158] A geometrically local commuting-projector code that realizes beyond-group-cohomology invertible topological phases in arbitrary dimensions. Instances of the code in 4D realize the 3D \(\mathbb{Z}_2\) gauge theory with fermionic charge and either bosonic (FcBl) or fermionic (FcFl) loop excitations at their boundaries [18,159]; see Ref. [160] for a different lattice-model formulation of the FcBl boundary code.
- Chien-Choy generalized BCH (GBCH) code[161] An \([n,k\geq n-rm, d\geq r+1]_q\) alternant code defined using two polynomials \(P(x),G(x)\) that are relatively prime to \(x^n-1\), with \(\deg P \leq n-1\) and \(r = \deg G \leq n-1\).
- Chiral semion Walker-Wang model code[162] A 3D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) whose low-energy excitations on boundaries realize the chiral semion topological order. The model admits 2D chiral semion topological order at one of its surfaces [162,163]. The corresponding phase can also be realized via a non-stabilizer Hamiltonian [164].
- Chiral semion subsystem code[165] Modular-qudit subsystem stabilizer code with qudit dimension \(q=4\) that is characterized by the chiral semion topological phase. Admits a set of geometrically local stabilizer generators on a torus.
- Chuang-Leung-Yamamoto (CLY) code[58] Bosonic Fock-state code that encodes \(k\) qubits into \(n\) oscillators, with each oscillator restricted to having at most \(N\) excitations. Codewords are superpositions of oscillator Fock states which have exactly \(N\) total excitations, and are either uniform (i.e., balanced) superpositions or unbalanced superpositions.
- Circuit-to-Hamiltonian approximate code[166] Approximate qubit block code that forms the ground-state space of a frustration-free Hamiltonian with non-commuting terms. Its distance and logical-qubit number are both of order \(\Omega(n/\log^5 n)\) [166; Thm. 3.1]. The code is an approximate non-stabilizer QLWC code since the Hamiltonian consists of non-commuting weight-ten non-Pauli projectors, with each qubit acted on by order \(O(\text{polylog}(n)\) projectors.
- Classical fractal liquid code[167,168] Member of a family of \([L^D,O(L^{D-1}),O(L^{D-\epsilon})]_p\) linear codes on \(D\)-dimensional square lattices of side length \(L\) and for some prime \(p\) and \(\epsilon > 0\) that is based on \(p\)-ary generalizations of the Sierpinski triangle.
- Classical topological code[169–171] Classical code defined on a two-dimensional lattice and derived from a geometrically local stabilizer code, such as the surface code or color code.
- Classical-product code[172–174] A CSS code constructed by separately constructing the \(X\) and \(Z\) check matrices using product constructions from classical codes. A particular \([[512,174,8]]\) code performed well [173] against erasure and depolarizing noise when compared to other notable CSS codes, such as the asymptotically good quantum Tanner codes. These codes have been generalized to the intersecting subset code family [174].
- Classical-quantum (c-q) code Code designed specifically for transmission of classical information through non-classical channels, e.g., quantum channels, hybrid quantum-classical channels, or channels with classical inputs and quantum outputs. Such codes include maps from a classical alphabet into a quantum Hilbert space.
- Clifford group-representation QSC[175] QSC whose projection is onto a copy of an irreducible representation of the single-qubit Clifford group, taken as the binary octahedral subgroup of the group \(SU(2)\) of Gaussian rotations. Its codewords consist of non-uniform superpositions of 48 coherent states.
- Clifford spin code[176,177] A single-spin code designed to realize a discrete group of gates using \(SU(2)\) rotations. Codewords are subspaces of a spin's Hilbert space that house irreducible representations (irreps) of a discrete subgroup of \(SU(2)\).
- Clifford subgroup-orbit QSC[178] A \(((2^r,2,2-\sqrt{2},8))\) QSC for \(r \geq 2\) constructed using the real-Clifford subgroup-orbit code. Logical constellations are constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group [179] to \(2\) different vectors on the complex sphere. The code is known as the Witting code for \(r=2\) because its two logical constellations form vertices of Witting polytopes.
- Clifford-deformed surface code (CDSC)[180] A generally non-CSS derivative of the surface code defined by applying a constant-depth Clifford circuit to the original (CSS) surface code. Unlike the surface code, CDSCs include codes whose thresholds and subthreshold performance are enhanced under noise biased towards dephasing. Examples of CDSCs include the XY code, XZZX code, and random CDSCs.
- Cluster-state code[181] a.k.a. Graph-state code.A code based on a cluster state and often used in measurement-based quantum computation (MBQC) [182] (a.k.a. one-way quantum processing), which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension. This is done by encoding the computation into the features of the cluster state''s graph.
- Code in permutations[183,184] a.k.a. Permutation-based code.Encodes codewords into permutations of \(n\) objects.
- Code with locality A code with \((r,\delta)\) locality is a code that encodes each codeword coordinate into an \([r+\delta-1,r,\delta]\) MDS code [185; Sec. 31.3.4.5]. In other words, given a codeword \(c\) and coordinate \(c_i\), there exists a coordinate set \(S_i\) of size \(\leq r+\delta-1\) such that the restriction \(C_{|S_i}\) of the code to that set is a code with minimum distance \(\delta\).
- Codeword stabilized (CWS) code[186,187] A code defined using a cluster state and a set of \(Z\)-type Pauli strings defined by a binary classical code.
- Coherent FSK (CFSK) c-q code[188,189] Coherent-state c-q code encoding into coherent states that are frequency-shifted with certain initial relative phase.
- Coherent-parity-check (CPC) code[190–192]
- Coherent-state c-q code Bosonic c-q code whose codewords form a constellation of coherent states. Encodes real numbers into coherent states for transmission over a quantum channel and decoding with a quantum-enhanced receiver.
- Coherent-state constellation code Qudit-into-oscillator code whose codewords can succinctly be expressed as superpositions of a countable set of coherent states that is called a constellation. Some useful constellations form a group (see gkp, cat or \(2T\)-qutrit codes) while others make up a Gaussian quadrature rule [195,196].
- Color code[7,8] Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs.
- Combinatorial PI code[197] A member of a family of PI quantum codes whose correction properties are derived from solving a family of combinatorial identities. The code encodes one logical qubit in superpositions of Dicke states whose coefficients are square roots of ratios of binomial coefficients.
- Combinatorial design a.k.a. Block design, Covering design.A constant-weight binary code that is mapped into a combinatorial \(t\)-design.
- Commuting-projector Hamiltonian code Hamiltonian-based code whose Hamiltonian terms can be expressed as orthogonal projectors (i.e., Hermitian operators with eigenvalues 0 or 1) that commute with each other.
- Compass code[198]
- Complete-intersection RM-type code[202] Evaluation code of polynomials evaluated on points lying on a complete intersection.
- Completely regular code[203] A code \(C\) is completely regular if the weight distribution of any coset \(e+C\) depends only on the distance \(d(e,C)\) of \(e\) to \(C\) [204].
- Complex Hadamard spherical code[205] A spherical code obtained from particular complex Hadamard matrices [206].
- Concatenated GKP code[207] A concatenated code whose outer code is a GKP code. In other words, a bosonic code that can be thought of as a concatenation of an arbitrary inner code and another bosonic outer code. Most examples encode physical qubits of an inner stabilizer code into the square-lattice GKP code.
- Concatenated Steane code[208,209] A member of the family of \([[7^m,1,3^m]]\) CSS codes, each of which is a recursive level-\(m\) concatenatenation of the Steane code. This family is one of the first to admit a concatenated threshold [208–212].
- Concatenated bosonic code A concatenated code whose outer code is a bosonic code. In other words, a bosonic code that can be thought of as a concatenation of a possibly non-bosonic inner code and another bosonic outer code.
- Concatenated c-q code A c-q code constructed out of two classical or quantum codes for the purposes of transmission of classical information over quantum channels.
- Concatenated cat code[213] A concatenated code whose outer code is a cat code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another cat outer code. Most examples encode physical qubits of an inner stabilizer code into the two-component cat code.
- Concatenated code[214] a.k.a. Serially concatenated code.A code whose encoding mapping is a composition of two mappings: first the message set is mapped onto the code space of the outer code, then each coordinate of the outer code is mapped on the code space of the inner code. In the basic construction, the outer code's alphabet is the finite field \(GF(p^m)\) and the \(m\)-dimensional inner code is over over the field \(GF(p)\). The construction is not limited to linear codes.
- Concatenated quantum code[215] A combination of two quantum codes, an inner code \(C\) and an outer code \(C^\prime\), where the physical subspace used for the inner code consists of the logical subspace of the outer code. In other words, first one encodes in the inner code \(C^\prime\), and then one encodes each of the physical registers of \(C^\prime\) in an outer code \(C\). An inner \(C = ((n_1,K,d_1))_{q_1}\) and outer \(C^\prime = ((n_2,q_1,d_2))_{q_2}\) block quantum code yield an \(((n_1 n_2, K, d \geq d_1d_2))_{q_2}\) concatenated block quantum code [216].
- Concatenated qubit code A concatenated code whose outer code is a qubit code. In other words, a qubit code that can be thought of as a concatenation of an arbitrary inner code and another qubit outer code. An inner \(C = ((n_1,K,d_1))\) and outer \(C^\prime = ((n_2,2,d_2))\) qubit code yield an \(((n_1 n_2, K, d \geq d_1d_2))\) concatenated qubit code.
- Conference code[217][56; pg. 55] A member of the family of \((n,2n+2,(n-1)/2)\) nonlinear binary codes for \(n=1\) modulo 4 that are constructed from conference matrices.
- Conformal-field theory (CFT) code[218,219]
- Constacyclic code a.k.a. Twisted code.A classical code \(C\) of length \(n\) over an alphabet \(R\) is \(\alpha\)-constacyclic (or \(\alpha\)-twisted) if, for each string \(c_1 c_2 \cdots c_n\in C\), the string \(\alpha c_n, c_1, \cdots, c_{n-1} \in C\). A \(-1\)-constacyclic code is called negacyclic.
- Constant-energy code Code whose codewords are points on a real or complex sphere whose radius squared is called the energy. Typically, only angular distances between points are relevant for code performance, so one can normalize codewords of a constant-energy code to obtain up a spherical code, i.e., a constant energy code with energy one.
- Constant-excitation (CE) code[220–222] Code whose codewords lie in an excited-state eigenspace of a Hamiltonian governing the total energy or total number of excitations of the underlying quantum system. For qubit codes, such a Hamiltonian is often the total spin Hamiltonian, \(H=\sum_i Z_i\). For spin-\(S\) codes, this generalizes to \(H=\sum_i J_z^{(i)}\), where \(J_z\) is the spin-\(S\) \(Z\)-operator. For bosonic codes, such as Fock-state codes, codewords are often in an eigenspace with eigenvalue \(N>0\) of the total excitation or energy Hamiltonian, \(H=\sum_i \hat{n}_i\).
- Constant-weight code A block code over a field or a ring whose codewords all have the same Hamming weight \(w\). The complement of a binary constant-weight code is a constant-weight code obtained by interchanging zeroes and ones in the codewords. The set of all binary codewords of length \(n\) forms the Johnson space \(J(n,w)\) [223–226].
- Constantin-Rao (CR) code[227] A nonlinear single-asymmetric-error code that generalize VT codes and that is constructed from an Abelian group.
- Construction-\(A\) code[228] a.k.a. Mod-two lattice code.
- Convolutional code[229] Infinite-block code that is formed using generator polynomials over the finite field with two elements. The encoder slides across contiguous subsets of the input bit-string (like a convolutional neural network) evaluating the polynomials on that window to obtain a number of parity bits. These parity bits are the encoded information.
- Covariant block quantum code[230] a.k.a. Equivariant block quantum code.A block code on \(n\) subsystems that admits a group \(G\) of transversal gates. The group has to be finite for finite-dimensional codes due to the Eastin-Knill theorem. Continuous-\(G\) covariant codes, necessarily infinite-dimensional, are relevant to error correction of quantum reference frames [230] and error-corrected parameter estimation.
- Covering code A \(q\)-ary code \(C\) is \(\rho\)-covering if \(\forall v \in GF(q)^{n}\), there is a codeword \(c \in C\) such that the Hamming distance \(D(c,v)\leq \rho\). More generally, a covering code in a metric space is covering if the union of balls of some radius centered at the codewords covers the entire space.
- Coxeter-Todd \(K_{12}\) lattice code[231] Even integral lattice in dimension \(12\) that exhibits optimal packing. It's automorphism group was discovered by Mitchell [232]. For more details, see [233][110; Sec. 4.9].
- Cross-interleaved RS (CIRS) code[234,235] An IRS code that is constructed using two shortened RS codes and two forms of interleaving. The code can also be visualized as a 2D array code [89].
- Crystalline-circuit qubit code[236] Code dynamically generated by unitary Clifford circuits defined on a lattice with some crystalline symmetry. A notable example is the circuit defined on a rotated square lattice with vertices corresponding to iSWAP gates and edges decorated by \(R_X[\pi/2]\), a single-qubit rotation by \(\pi/2\) around the \(X\)-axis. This circuit is invariant under space-time translations by a unit cell \((T, a)\) and all transformations of the square lattice point group \(D_4\).
- Cubeoctahedron code Spherical \((3,12,1)\) code whose codewords are the vertices of the cubeoctahedron. Codewords form the minimal lattice-shell code of the \(D_3\) face-centered cubic (fcc) lattice.
- Cubic honeycomb color code[8] 3D color code defined on a four-colorable bitruncated cubic honeycomb uniform tiling.
- Cubic theory code[237] a.k.a. Magic stabilizer code.A geometrically local commuting-projector code defined on triangulations of lattices in arbitrary spatial dimensions. Its code Hamiltonian terms include Pauli-\(Z\) operators and products of Pauli-\(X\) operators and \(CZ\) gates. The Hamiltonian realizes higher-form \(\mathbb{Z}_2^3\) gauge theories whose excitations obey non-Abelian Ising-like fusion rules.
- Cycle LDPC code[238] An LDPC code whose parity-check matrix forms the incidence matrix of a graph, i.e., has weight-two columns.
- Cycle code[169,238–242] a.k.a. Graph theoretic code, Graph homology code, Graph code.A code whose parity-check matrix forms the incidence matrix of a graph. This code's properties are derived from the size two chain complex associated with the graph.
- Cyclic code[243–247] A code of length \(n\) over an alphabet is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword.
- Cyclic linear \(q\)-ary code A \(q\)-ary code of length \(n\) is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword. A cyclic code is called primitive when \(n=q^r-1\) for some \(r\geq 2\). A shortened cyclic code is obtained from a cyclic code by taking only codewords with the first \(j\) zero entries, and deleting those zeroes.
- Cyclic linear binary code A binary code of length \(n\) is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword. A cyclic code is called primitive when \(n=2^r-1\) for some \(r\geq 2\). A shortened cyclic code is obtained from a cyclic code by taking only codewords with the first \(j\) zero entries, and deleting those zeroes.
- Cyclic quantum code[248] A block quantum code such that cyclic permutations of the subsystems leave the codespace invariant. In other words, the automorphism group of the code contains the cyclic group \(\mathbb{Z}_n\).
- Cyclic redundancy check (CRC) code[247,249,250] a.k.a. Frame check sequence (FCS).A generalization of the single parity-check code in which the generalization of the parity bit is the remainder of the data string (mapped into a polynomial via the Cyclic-to-polynomial correspondence) divided by some generator polynomial. A notable family of codes is referred to as CRC-(\(m-1\)), where \(m\) is the length of the generator polynomial.
- DNA storage code[251] Code that was designed (or that can be applied) to encode information into the four-base-pair alphabet of a DNA molecule.
- Deligne-Lusztig code[252–255] Evaluation code of polynomials evaluated on points lying on a Deligne-Lusztig curve.
- Delsarte-Goethals (DG) code[256] Member of a family of \((2^{2t+2},2^{(2t+1)(t-d+2)+2t+3},2^{2t+1}-2^{2t+1-d})\) binary nonlinear codes for parameters \(r \geq 1\) and \(m = 2t+2 \geq 4\), denoted by DG\((m,r)\), that generalizes the Kerdock code.
- Denniston code[257] Projective code that is part of a family of \([2^{a+i}+2^i-2^a,3,2^{a+i}-2^a]_{GF(2^a)}\) codes for \(i < a\) constructed using Denniston arcs.
- Derby-Klassen (DK) code[258,259] a.k.a. Compact encoding.A fermion-into-qubit code defined on regular tilings with maximum degree 4 whose stabilizers are associated with loops in the tiling. The code outperforms several other encodings in terms of encoding rate [258; Table I]. It has been extended for models with several modes per site [260].
- Determinant code[261] Determinant codes give optimal exact repair regenerating codes for any \([n,k,d=k]\) at all the points of the storage bandwidth trade-off curve. The codes are linear, and the exact regenerating property is provided based on fundamental properties of matrix determinants. The field size \(q\) required for this code construction is linear in \(n\).
- Diagonal code[262] Member of an explicit family of high-rate \([n,k,d,\alpha, \beta = \frac{\alpha}{d-k+1}, M=k\alpha]\) MSR codes for any \(r\) and \(n\). Such codes can optimally repair any \(f\) failed nodes from any \(d\) helper nodes for all \(d\), \(1 \le f \le r\) and \(k \le d \le n-f\) simultaneously. These codes can be constructed over any base field \(GF(q)\) as long as \(|GF(q)| \ge sn\), where \(s = \text{lcm}(1,2,\cdots,r)\).
- Diatomic molecular code[263; Sec. VI] Approximate quantum code that encodes a qudit in the finite-dimensional Hilbert space of a rigid body with \(SO(2)\) symmetry (e.g., a heteronuclear diatomic molecule). This state space is the space of normalized functions on the two-sphere, consisting of a direct sum of all non-negative integer angular momenta. Ideal codewords may not be normalizable because the space is infinite-dimensional, so approximate versions have to be constructed in practice.
- Difference-set cyclic (DSC) code[264] Cyclic LDPC code constructed deterministically from a difference set. Certain DCS codes satisfy more redundant constraints than Gallager codes and thus can outperform them [265].
- Dihedral \(G=D_m\) quantum-double code[35,266] Quantum-double code whose codewords realize \(G=D_m\) topological order associated with a \(2m\)-element dihedral group \(D_m\). Includes the simplest non-Abelian order \(D_3 = S_3\) associated with the permutation group of three objects. The code can be realized as the ground-state subspace of the quantum double model, defined for \(D_m\)-valued qudits [35]. An alternative qubit-based formulation realizes the gauged \(G=\mathbb{Z}_3^2\) twisted quantum double phase [266], which is the same topological order as the \(G=D_4\) quantum double [267,268].
- Dijkgraaf-Witten gauge theory code[269–271] A code whose codewords realize \(D\)-dimensional lattice Dijkgraaf-Witten gauge theory [269,270] for a finite group \(G\) and a \(D+1\)-cocycle \(\omega\) in the cohomology class \(H^{D+1}(G,U(1))\). When the cocycle is non-trivial, the gauge theory is called a twisted gauge theory. For trivial cocycles in 3D, the model can be called a quantum triple model, in allusion to being a 3D version of the quantum double model. There exist lattice-model formulations in arbitrary spatial dimension [271] as well as explicitly in 3D [272,273].
- Dinur code[274] Member of infinite family of locally testable \([n,n/\text{polylog}(n),d]\) codes with vanishing rate. Code construction relies on a construction utilizing tensor-product codes [275].
- Dinur-Hsieh-Lin-Vidick (DHLV) code[276] A family of asymptotically good QLDPC codes which are related to expander LP codes in that the roles of the check operators and physical qubits are exchanged.
- Dinur-Lin-Vidick (DLV) code[277] Member of a family of quantum locally testable codes constructed using cubical chain complexes, which are \(t\)-order extensions of the complexes underlying expander codes (\(t=1\)) and expander lifted-product codes (\(t=2\)).
- Disphenoidal 288-cell code Spherical \((4,48,2-\sqrt{2})\) code [116; Ex. 1.2.6] whose codewords are the vertices of the disphenoidal 288-cell. Codewords are the union of two 24-point lattice shells of the \(D_4\) lattice. The first shell consists of the 24 permutations of the four vectors \((0,0,\pm 1,\pm 1)\), and the second of the 16 vectors \((\pm 1,\pm 1,\pm 1,\pm 1)\) and the 8 permutations of the vectors \((0,0,0,\pm 2)\). A realization in terms of quaternion coordinates yields the 48 elements of the binary octahedral group \(2O\) [27; Sec. 8.6].
- Distance-balanced code[278,279] Galois-qudit CSS code constructed from a CSS code and a classical code using a distance-balancing procedure based on a generalized homological product. The initial code is said to be unbalanced, i.e., tailored to noise biased toward either bit- or phase-flip errors, and the procedure can result in a code that is treats both types of errors on a more equal footing. The original distance-balancing procedure [278], later generalized [279; Thm. 4.2], can yield QLDPC codes [278; Thm. 1].
- Distributed computation code Encoding that provides an extra redundancy for distributed matrix computation algorithms such as matrix multiplication. Parallelized algorithms distribute a desired computation over many nodes, and a key performance bottleneck is due to some nodes completing their individual tasks much later than other nodes. Matrix computation codes provide a layer of redundancy such that the computation can be performed without having all nodes finish their piece of the computation.
- Distributed-storage code Block code designed to encode information into spatial nodes such that it is possible to recover said information after failure of some helper nodes by accessing the remaining nodes with minimal bandwidth.
- Divisible code[280] A linear \(q\)-ary block code is \(\Delta\)-divisible if the Hamming weight of each of its codewords is divisible by divisor \(\Delta\). A \(2\)-divisible (\(4\)-divisible, \(8\)-divisible) code is called even (doubly even, triply even) [110,281]. A code is called singly-even if all codewords are even and at least one has weight equal to 2 modulo 4. More generally, a code is \(m\)-even if it is \(2^{m}\)-divisible.
- Dodecacode[282] The unique trace-Hermitian self-dual additive \((12,4^6,6)_4\) code. Its codewords are cyclic permutations of \((\omega 10100100101)\), where \(GF(4)=\{0,1,\omega,\bar{\omega}\}\) is the quaternary Galois field [283; Sec. 2.4.8]. Another generator matrix can be found in [284; Ex. 9.10.8].
- Double-semion stabilizer code[33,285] A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) that is characterized by the 2D double semion topological phase. The code can be obtained from the \(\mathbb{Z}_4\) surface code by condensing the anyon \(e^2 m^2\) [165]. Originally formulated as the ground-state space of a Hamiltonian with non-commuting terms [285], which can be extended to other spatial dimensions [286], and later as a commuting-projector code [34,287].
- Doubled color code[288–290] Family of \([[2t^3+8t^2+6t-1,1,2t+1]]\) subsystem color codes (with \(t\geq 1\)), constructed using a generalization of the doubling transformation [291], that admit a Clifford + \(T\) transversal gate set using gauge fixing.
- Dual additive code For any \(q\)-ary additive code \(C\), the dual additive (or orthogonal additive) code is \begin{align} C^\perp = \{ y\in GF(q)^{n} ~|~ x \star y=0 \forall x\in C\}, \tag*{(3)}\end{align} where the trace inner product is \(x\star y = \sum_{i=1}^n \text{tr}(x_i y_i)\) for coordinates \(x_i,y_i\) and for \(\textit{tr}\) being the field trace.
- Dual code over \(R\) For any linear code \(C\) over a ring \(R\), the dual code over \(R\) is \begin{align} C^\perp = \{ y\in R^{n} ~|~ x \cdot y=0 \forall x\in C\}, \tag*{(4)}\end{align} where the ordinary, standard, or Euclidean inner product is \(x\cdot y = \sum_{i=1}^n x_i y_i\) for coordinates \(x_i,y_i\).
- Dual lattice code a.k.a. Reciprocal lattice code, Polar lattice code.For any dimensional lattice \(L\), the dual lattice is the set of vectors whose inner products with the elements of \(L\) are integers.
- Dual linear code a.k.a. Orthogonal linear code.For any \([n,k]_q\) linear code \(C\), the dual code is the set of \(q\)-ary strings that are orthogonal to the codewords of \(C\) under a particular inner product.
- Dual polytope code For any spherical code whose codewords are vertices of a polytope, the dual code consists of codewords that are centers of the faces of said polytope. The dual codewords make up the vertices of the polytope dual to the original polytope.
- Dual-rail quantum code[292] Two-mode bosonic code encoding a logical qubit in Fock states with one excitation. The logical-zero state is represented by \(|10\rangle\), while the logical-one state is represented by \(|01\rangle\). This encoding is often realized in temporal or spatial modes, corresponding to a time-bin or frequency-bin encoding. Two different types of photon polarization can also be used.
- Dynamical automorphism (DA) code[293,294] a.k.a. Dynamical code, Aperiodic Floquet code.Dynamically-generated stabilizer-based code whose (not necessarily periodic) sequence of few-body measurements implements state initialization, logical gates and error detection.
- Dynamically-generated QECC[295] Block quantum code whose natural definition is in terms of a many-body scaling limit of a local dynamical process. Such processes, which are often non-deterministic, update the code structure and can include random unitary evolution or non-commuting projective measurements.
- EA FG-QLDPC code[296] One of several EA QLDPC code families constructed from finite-geometry LDPC (FG-LDPC) codes. There exists a family that requires an amount of entanglement that vanishes linearly with the length of each code.
- EA Galois-qudit code Galois-qudit code designed to utilize pre-shared entanglement between sender and receiver.
- EA Galois-qudit stabilizer code[297] A Galois-qudit stabilizer code constructed using a variation of the stabilizer formalism designed to utilize pre-shared entanglement between sender and receiver. A code is typically denoted as \([[n,k;e]]_q\) or \([[n,k,d;e]]_q\), where \(d\) is the distance of the underlying non-EA \([[n,k,d]]_q\) code, and \(e\) is the number of required pre-shared maximally entangled Galois-qudit maximally entangled states.
- EA MDS code[298–300] EA Galois-qudit code whose parameters make the EAQECC Singleton bound (a.k.a. qubit-ebit Singleton bound) [300; Thm. 6] become an equality.
- EA QC-QLDPC code[296] One of several EA QLDPC code families constructed from QC-LDPC codes.
- EA QLDPC code EA qubit stabilizer code for which the number of sites participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant \(w\) as \(n\to\infty\)
- EA analog stabilizer code[301] Constructed using a variation of the analog stabilizer formalism designed to utilize pre-shared entanglement between sender and receiver.
- EA bosonic code Bosonic code designed to utilize pre-shared entanglement between sender and receiver.
- EA combinatorial-design QLDPC code[302] One of several EA QLDPC code families constructed from combinatorial designs.
- EA quantum LCD code[303] An EA Galois-qudit stabilizer code constructed from an LCD code. This family include the first asymptotically good EA Galois-qudit codes.
- EA quantum convolutional code[304–306] A quantum convolutional code designed to utilize pre-shared entanglement between sender and receiver, which can reduce memory requirements [307].'
- EA quantum turbo code[308,309] A quantum turbo code which uses pre-shared entanglement. This allows its encoder to be both recursive and non-catastrophic.
- EA qubit code Qubit code designed to utilize pre-shared entanglement between sender and receiver.
- EA qubit stabilizer code[298,310] Constructed using a variation of the stabilizer formalism designed to utilize pre-shared entanglement between sender and receiver. A code is typically denoted as \([[n,k;e]]\) or \([[n,k,d;e]]\), where \(d\) is the distance of the underlying non-EA \([[n,k,d]]\) code, and \(e\) is the number of required pre-shared maximally entangled Bell states (ebits). While other entangled states can be used, there is always a choice a generators such that the Bell state suffices while still using the fewest ebits.
- EVENODD code[311] A binary array code that can correct any two disk failures (i.e., two erasures). See [89] for more details.
- Editing code[312] a.k.a. Insertion and deletion code.A code designed to protect against insertions, where a new symbol is added somewhere within the string, and deletions, where a symbol at an unknown location is erased.
- Eigenstate thermalization hypothesis (ETH) code[313] a.k.a. Thermodynamic code.An \(n\)-qubit approximate code whose codespace is formed by eigenstates of a translationally-invariant quantum many-body system which satisfies the Eigenstate Thermalization Hypothesis (ETH). ETH ensures that codewords cannot be locally distinguished in the thermodynamic limit. Relevant many-body systems include 1D non-interacting spin chains or frustration-free systems such as Motzkin chains and Heisenberg models.
- Elliptic code Evaluation AG code of rational functions evaluated on points lying on an elliptic curve, i.e., a curve of genus one.
- Entanglement-assisted (EA) QECC[298,310,314] a.k.a. Catalytic QECC.QECC whose encoding and decoding utilizes pre-shared entanglement between sender and receiver.
- Entanglement-assisted (EA) hybrid quantum code[315–317] Code that encodes quantum and classical information and requires pre-shared entanglement for transmission.
- Entanglement-assisted (EA) subsystem QECC[318,319] a.k.a. EA operator QECC.Subsystem QECC whose encoding and decoding utilizes pre-shared entanglement between sender and receiver.
- Error-corrected sensing code[320,321] Code that can be obtained via an optimization procedure that ensures correction against a set \(\cal{E}\) of errors as well as guaranteeting optimal precision in locally estimating a parameter using a noiseless ancilla. For tensor-product spaces consisting of \(n\) subsystems (e.g., qubits, modular qudits, or Galois qudits), the procedure can yield a code whose parameter estimation precision satisfies Heisenberg scaling, i.e., scales quadratically with the number \(n\) of subsystems.
- Error-correcting code (ECC) Code designed for transmission of classical information through classical channels in a robust way.
- Error-correcting output code (ECOC)[322,323] A length-\(n\) binary or ternary (over \(\{\pm 1,0\}\)) block code used to convey information about classes to classifiers in multiclass machine learning. Rows of the code's generator matrix denote different classes, while columns correspond to classifiers.
- Evaluation AG code Evaluation code over \(GF(q)\) on a set of points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) in \(GF(q)\) lying on an algebraic curve \(\cal X\) whose corresponding vector space \(L\) of functions \(f\) consists of certain polynomials or rational functions.
- Evaluation code[324–326] Code whose codewords are evaluations of functions at certain fixed points. Code properties can be inferred from the structure of the functions and the underlying geometric object containing the points, often using results from algebraic geometry.
- Expander LP code[327] Family of \(G\)-lifted product codes constructed using two random classical Tanner codes defined on expander graphs [328]. For certain parameters, this construction yields the first asymptotically good QLDPC codes. Classical codes resulting from this construction are one of the first two families of \(c^3\)-LTCs.
- Expander code[329] LDPC code whose parity-check matrix is derived from the adjacency matrix of bipartite expander graph [328] such as a Ramanujan graph or a Cayley graph of a projective special linear group over a finite field [330,331]. Expander codes admit efficient encoding and decoding algorithms and yield an explicit (i.e., non-random) asymptotically good LDPC code family.
- Extended GRS code A GRS code with an additional parity-check coordinate with corresponding evaluation point of zero. In other words, an \([n+1,k,n-k+2]_q\) GRS code whose polynomials are evaluated at the points \((\alpha_1,\cdots,\alpha_n,0)\). The case when \(n=q-1\), multipliers \(v_i=1\), and \(\alpha_i\) are \(i-1\)st powers of a primitive \(n\)th root of unity is an extended narrow-sense RS code.
- Extended IRA (eIRA) code[332–334] A generalization of the IRA code in which the outer LDGM code is replaced by a random sparse matrix containing no weight-two columns.
- Fermion code Finite-dimensional quantum error-correcting code encoding a logical (qudit or fermionic) Hilbert space into a physical Fock space of fermionic modes. Codes are typically described using Majorana operators, which are linear combinations of fermionic creation and annihilation operators [141].
- Fermion-into-qubit code Qubit stabilizer code encoding a logical fermionic Hilbert space into a physical space of \(n\) qubits. Such codes are primarily intended for simulating fermionic systems on quantum computers, and some of them have error-detecting, correcting, and transmuting properties.
- Fiber-bundle code[335] a.k.a. Twisted product code.A CSS code constructed by combining one code as the base and another as the fiber of a fiber bundle. In particular, taking a random LDPC code as the base and a cyclic repetition code as the fiber yields, after distance balancing, a QLDPC code with distance of order \(\Omega(n^{3/5}\text{polylog}(n))\) and rate of order \(\Omega(n^{-2/5}\text{polylog}(n))\) is obtained.
- Fibonacci code[168] The code is defined on an \(L\times L/2\) lattice with one bit on each site, where \(L=2^N\) for an integer \(N\geq 2\). The codewords are defined to satisfy the condition that, for each lattice site \((x,y)\), the bits on \((x,y)\), \((x+1,y)\), \((x-1,y)\) and \((x,y+1)\) (where the lattice is taken to be periodic in both directions) contain an even numbers of \(1\)'s. The codewords can be generated using a one-dimensional cellular automaton of length \(L\) (periodic). The \(2^L\) possible initial states correspond to the \(2^L\) codewords. For each generation, the state of each cell is the xor sum of that cell and its two neighbors in the previous generation. After \(L/2-1\) generations, the entire history generated by the automaton corresponds to a codeword, where the initial state is the first row of the lattice, the first generation is the second row, etc.
- Fibonacci fractal spin-liquid code[168] A fractal type-I fracton CSS code defined on a cubic lattice [19; Eq. (D23)].
- Fibonacci string-net code[285,336] Quantum error correcting code associated with the Levin-Wen string-net model with the Fibonacci input category, admitting two types of encodings.
- Finite-dimensional error-correcting code (ECC)[337] An error-correcting code defined over a finite alphabet.
- Finite-dimensional quantum error-correcting code Encodes quantum information in a \(K\)-dimensional (logical) subspace of an \(N\)-dimensional (physical) Hilbert space such that it is possible to recover said information from errors. The logical subspace is spanned by a basis comprised of code basis states or codewords.
- Finite-geometry (FG) QLDPC code[338–340]
- Finite-geometry LDPC (FG-LDPC) code[341] LDPC code whose parity-check matrix is the incidence matrix of points and hyperplanes in either a Euclidean or a projective geometry. Such codes are called Euclidean-geometry LDPC (EG-LDPC) and projective-geometry LDPC (PG-LDPC), respectively. Such constructions have been generalized to incidence matrices of hyperplanes of different dimensions [342].
- Five-qubit perfect code[343,344] a.k.a. Laflamme code.Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error.
- Five-rotor code[345] Extension of the five-qubit stabilizer code to the integer alphabet, i.e., the angular momentum states of a planar rotor. The code is \(U(1)\)-covariant and ideal codewords are not normalizable.
- Flag-variety code[346] Evaluation code of polynomials evaluated on points lying on a flag variety.
- Floquet color code[347–349] a.k.a. CSS Floquet toric code, \(\mathbb{Z}_2\) Floquet code, CSS honeycomb code.Floquet code on a trivalent 2D lattice whose parent topological phase is the \(\mathbb{Z}_2\times\mathbb{Z}_2\) 2D color-code phase and whose measurements cycle logical quantum information between the nine \(\mathbb{Z}_2\) surface-code condensed phases of the parent phase. The code's ISG is the stabilizer group of one of the nine surface codes.
- Fock-state bosonic code Qudit-into-oscillator code whose protection against AD noise (i.e., photon loss) stems from the use of disjoint sets of Fock states for the construction of each code basis state. The simplest example is the dual-rail code, which has codewords consisting of single Fock states \(|10\rangle\) and \(|01\rangle\). This code can detect a single loss error since a loss operator in either mode maps one of the codewords to a different Fock state \(|00\rangle\). More involved codewords consist of several well-separated Fock states such that multiple loss events can be detected and corrected.
- Folded RS (FRS) code[350] A linear \([n/m,k]_{q^m}\) code that is a modification of an \([n,k]_q\) RS code such that evaluations are grouped to yield a code with smaller length. In this case, the evaluation points are all powers of a generating field element \(\gamma\), \(\alpha_i=\gamma^i\). Each codeword \(\mu\) of an \(m\)-folded RS code is a string of \(n/m\) symbols, with each symbol being a string of values of a polynomial \(f_\mu\) at consecutive powers of \(\gamma\), \begin{align} \begin{split} \mu\to&\Big(\left(f_{\mu}(\alpha^{0}),\cdots,f_{\mu}(\alpha^{m-1})\right),\left(f_{\mu}(\alpha^{m}),\cdots,f_{\mu}(\alpha^{2m-1})\right)\cdots\\&\cdots,\left(f_{\mu}(\alpha^{n-m}),\cdots,f_{\mu}(\alpha^{n-1})\right)\Big)~. \end{split} \tag*{(5)}\end{align}
- Folded quantum RS (FQRS) code[351] CSS code on \(q^m\)-dimensional Galois-qudits that is constructed from folded RS (FRS) codes (i.e., an RS code whose coordinates have been grouped together) via the Galois-qudit CSS construction. This code is used to construct Singleton-bound approaching approximate quantum codes.
- Fountain code[352] Code based on the idea of generating an endless stream of custom encoded packets for the receiver. The code is designed so that the receiver can recover the original transmission of size \(Kl\) bits after receiving at least \(K\) packets each of \(l\) bits.
- Four-qubit single-deletion code[353,354] Four-qubit PI code that is the smallest qubit code to correct one deletion error.
- Four-rotor code[345; Sec. VIII] \([[4,2,2]]_{\mathbb Z}\) CSS rotor code that is an extension of the four-qubit code to the integer alphabet, i.e., the angular momentum states of a planar rotor.
- Fractal surface code[355–357] Kitaev surface code on a fractal geometry, which is obtained by removing qubits from the surface code on a cubic lattice. A related construction, the fractal product code, is a hypergraph product of two classical codes defined on a Sierpinski carpet graph [355]. The underlying classical codes form classical self-correcting memories [358–360].
- Fracton Floquet code[348] Floquet code whose qubits are placed on vertices of a truncated cubic honeycomb. Its weight-two check operators are placed on edges of each truncated cube, while weight-three check operators are placed on each triangle. Its ISG can be that of the X-cube model code or the checkerboard model code. On a three-torus of size \(L_x \times L_y \times L_z\), the code consists of \(n= 48L_xL_yL_z\) physical qubits and encodes \(k= 2(L_x+L_y+L_z)-6\) logical qubits.
- Fracton stabilizer code[361] A 3D translationally invariant modular-qudit stabilizer code whose codewords make up the ground-state space of a Hamiltonian in a fracton phase. Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted.
- Frameproof (FP) code[362,363] A block code designed to prevent a group of users from framing another user outside of the group for creating an unauthorized copy of data. FP codes help to provide software protection from the illegal distribution and copying of computer software and copyrighted materials. These codes help protect products of distributors as well as other naive users from being framed for illegal activity [364].
- Freedman-Meyer-Luo code[365] Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [366]. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries [367]. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology.
- Frequency-shift keying (FSK) code A \(q\)-ary frequency-shift keying (\(q\)-FSK) encodes one \(q\)-ary digit of information into signals with \(q\) different frequencies.
- Frobenius code[368] A cyclic prime-qudit stabilizer code whose length \(n\) divides \(p^t + 1\) for some positive integer \(t\).
- Frustration-free Hamiltonian code Hamiltonian-based code whose Hamiltonian is frustration free, i.e., whose ground states minimize the energy of each term.
- Fusion-based quantum computing (FBQC) code[369] Code whose codewords are resource states used in an FBQC scheme. Related to a cluster state via Hadamard transformations.
- GKP CV-cluster-state code[370] a.k.a. Hybrid cluster-state code.Cluster-state code can consists of a generalized analog cluster state that is initialized in GKP (resource) states for some of its physical modes. Alternatively, it can be thought of as an oscillator-into-oscillator GKP code whose encoding consists of initializing \(k\) modes in momentum states (or, in the normalizable case, squeezed vacua), \(n-k\) modes in (normalizable) GKP states, and applying a Gaussian circuit consisting of two-body \(e^{i V_{jk} \hat{x}_j \hat{x}_k }\) for some angles \(V_{jk}\). Provides a way to perform fault-tolerant MBQC, with the required number \(n-k\) of GKP-encoded physical modes determined by the particular protocol [370–373].
- GKP-surface code[372,374]
- GNU PI code[380,381] PI code whose codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of the binomial distribution.
- Gabidulin code[382–384] a.k.a. Vector rank-metric code, Delsarte-Gabidulin code.A linear code over \(GF(q^N)\) that corrects errors over rank metric instead of the traditional Hamming distance. Every element \(GF(q^N)\) can be written as an \(N\)-dimensional vector with coefficients in \(GF(q)\), and the rank of a set of elements is rank of the matrix formed by their coefficients.
- Gallager (GL) code[385,386] The first LDPC code. The rows of the parity check matrix of this regular code are divided into equal subsets, and columns in the first subset are randomly permuted to yield the corresponding rows in subsequent subsets.
- Galois-qudit BCH code[387–393] True Galois-qudit stabilizer code constructed from BCH codes via either the Hermitian construction or the Galois-qudit CSS construction. Parameters can be improved by applying Steane enlargement [394].
- Galois-qudit CSS code[395–401] a.k.a. Euclidean code.An \([[n,k,d]]_q \) Galois-qudit true stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Galois-qudit Pauli strings. Codes can be defined from chain complexes over \(GF(q)\) via an extension of qubit CSS-to-homology correspondence to Galois qudits.
- Galois-qudit CWS code A CWS code for Galois qudits, defined using a Galois-qudit cluster state and a set of Galois-qudit \(Z\)-type Pauli strings defined by a \(q\)-ary classical code.
- Galois-qudit GRS code[402,403]
- Galois-qudit RS code[402] a.k.a. Galois-qudit polynomial code (QPyC).An \([[n,k,n-k+1]]_q\) (with \(q>n\)) Galois-qudit CSS code constructed using two RS codes over \(GF(q)\).
- Galois-qudit USt code[407–411] a.k.a. Galois-qudit non-stabilizer code.A Galois-qubit code whose codespace consists of a direct sum of a Galois-qubit stabilizer codespace and one or more of that stabilizer code's error spaces.
- Galois-qudit code a.k.a. \(GF(q)\)-qudit code, \(\mathbb{F}_q\)-qudit code, Galois-qudit subspace code.Encodes \(K\)-dimensional Hilbert space into a \(q^n\)-dimensional (\(n\)-qudit) Hilbert space, with canonical qudit states \(|k\rangle\) labeled by elements \(k\) of the Galois field \(GF(q)\) and with \(q\) being a power of a prime \(p\).
- Galois-qudit color code[412] a.k.a. \(\mathbb{F}_q\)-qudit color code.Extension of the color code to 2D lattices of Galois qudits.
- Galois-qudit quantum RM code[413] True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via the Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [414; Sec. 4.2].
- Galois-qudit stabilizer code[415,416] An \(((n,K,d))_q\) Galois-qudit code whose logical subspace is the joint eigenspace of commuting Galois-qudit Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace, and the stabilizer group does not contain \(e^{i \phi} I\) for any \(\phi \neq 0\). The distance \(d\) is the minimum weight of a Galois-qudit Pauli string that implements a nontrivial logical operation in the code.
- Galois-qudit surface code[417,418] a.k.a. \(\mathbb{F}_q\)-qudit surface code.Extension of the surface code to 2D lattices of Galois qudits.
- Gauss' law code[419,420] An \([m+Dm,Dm,3]\) linear binary code for \(m\geq 3^D\), defined by the Gauss' law constraint of a \(D\)-dimensional fermionic \(\mathbb{Z}_2\) gauge theory [420; Thm. 1]. The code can be re-phrased as a distance-one stabilizer code whose stabilizers consist of gauge-group elements. It can be concatenated to form a stabilizer code for fault-tolerant quantum simulation of the underlying gauge theory [419,420].
- Generalized 2D color code[421] Member of a family of non-Abelian 2D topological codes, defined by a finite group \( G \), that serves as a generalization of the color code (for which \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)).
- Generalized EVENODD code[422] a.k.a. Blaum-Bruck-Vardy array code.
- Generalized Gallager code[423] A LDPC code that generalizes the Gallager codes using the Tanner construction. While Gallager code parity-check matrices consists of repetition code submatrices that are randomly permuted, generalized Gallager code matrices substitute general binary linear codes.
- Generalized RM (GRM) code[424–426] Reed-Muller code GRM\(_q(r,m)\) of length \(n=q^m\) over \(GF(q)\) with \(0\leq r\leq m(q-1)\). Its codewords are evaluations of the set of all degree-\(\leq r\) polynomials in \(m\) variables at the points of \(GF(q)\).
- Generalized RS (GRS) code An \([n,k,n-k+1]_q\) linear code that is a modification of the RS code where codeword polynomials are multiplied by additional prefactors.
- Generalized Shor code[92,427] Qubit CSS code constructed by concatenating two classical codes in a way the generalizes the Shor and quantum parity codes.
- Generalized Srivastava code[428] An \([n,k \geq n-mst,d \geq st+1 ]_q\) alternant code defined for \(n+s\) distinct elements \(\alpha_1,\alpha_2,\cdots,\alpha_n,w_1,w_2,\cdots,w_s\) and \(n\) nonzero elements \(z_1,z_2,\cdots,z_n\) of \(GF(q^m)\).
- Generalized bicycle (GB) code[429,430] a.k.a. Hyperbicycle code, Quasi-cyclic LP code.A quasi-cyclic Galois-qudit CSS code constructed using a generalized version of the bicycle ansatz [111] from a pair of equivalent index-two quasi-cyclic linear codes. Equivalently, the codes can constructed via the lifted-product construction for \(G\) being a cyclic group [29; Sec. III.E].
- Generalized concatenated code (GCC)[431,432] a.k.a. Cascade code.A code that combines multiple outer codes of the same length and (possibly) different dimensions with a single inner code; see Refs. [433][56; Ch. 18].
- Generalized five-squares code[434–436]
- Generalized homological-product CSS code CSS code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes.
- Generalized homological-product code Stabilizer code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes. The Qubit CSS-to-homology correspondence yields an interpretation of codes in terms of manifolds, thus allowing for the use of various products from topology in constructing codes.
- Generalized homological-product qubit CSS code Qubit CSS code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes.
- Generalized quantum Tanner code[437] An extension of quantum Tanner codes to codes constructed from two commuting regular graphs with the same vertex set. This allows for code construction using finite sets and Schreier graphs, yielding a broader family of square complexes.
- Generalized quantum divisible code[438] A level-\(\nu\) generalized quantum divisible code is a CSS code whose \(X\)-type stabilizers, in the symplectic representation, have zero norm and form a \((\nu,t)\)-null matrix (defined below) with respect to some odd-integer vector \(t\) [438; Def. V.1]. Such codes admit gates at the \(\nu\)th level of the Clifford hierarchy. Such codes can also be level-lifted [438; Theorem V.6], \(\nu\to\nu+1\), which recursively yields towers of generalized divisible codes from a particular ground code.
- Glynn code[439] The unique trace-Hermitian self-dual \([10,5,6]_9\) code, constructed using a 10-arc in \(PG(4,9)\) that is not a rational curve.
- Goethals code[440] Member of a family of \((2^m,2^{2^m-3m+1},8)\) binary nonlinear codes for \(m \geq 6\) that generalizes the Preparata codes. The code can be constructed as disjoint union of cosets of a certain linear code [56; Ch. 15].
- Golay code[441] A \([23, 12, 7]\) perfect binary linear code with connections to various areas of mathematics, e.g., lattices [110] and sporadic simple groups [56]. Adding a parity bit to the code results in the self-dual \([24, 12, 8]\) extended Golay code. Up to equivalence, both codes are unique for their respective parameters [442]. Shortening the Golay code yields the \([22,10,8]\), \([22,11,7]\), and \([22,12,6]\) shortened Golay codes [443]. The dual of the Golay code is its \([23,11,8]\) even-weight subcode [444,445].
- Gold code[446] Member of the family of \([2^r-1, 2r ]\) cyclic binary linear codes characterized by the generator polynomial of degree \(r\) of two maximum-period sequences of period \(2^r-1\) with absolute cross-correlation \( \leq 2^{(r+2)/2}\). Gold codewords are generated using \(m\)-sequences \(x\) and \(y\), which are codewords of simplex codes with check polynomials of degree \(r\) [446].
- Golden code[447] Variant of the Guth-Lubotzky hyperbolic surface code that uses regular tessellations for 4-dimensional hyperbolic space.
- Goldreich-Sudan code[448] Locally testable \([n,k,d]\) code with \(n = k^{1+O(1/u)}\) and distance of order \(\Omega(n)\) for query complexity \(u\). The same work also presented a probabilistic construction of codes of size \(k^{1+o(1)}\).
- Good QLDPC code Also called asymptotically good QLDPC codes. A family of QLDPC codes \([[n_i,k_i,d_i]]\) whose asymptotic rate \(\lim_{i\to\infty} k_i/n_i\) and asymptotic distance \(\lim_{i\to\infty} d_i/n_i\) are both positive.
- Goppa code[449–451] a.k.a. LG code.Let \( G(x) \) be a polynomial describing a projective-plane curve with coefficients from \( GF(q^m) \) for some fixed integer \(m\). Let \( L \) be a finite subset of the extension field \( GF(q^m) \) where \(q\) is prime, meaning \( L = \{\alpha_1, \cdots, \alpha_n\} \) is a subset of nonzero elements of \( GF(q^m) \). A Goppa code \( \Gamma(L,G) \) is an \([n,k,d]_q\) linear code consisting of all vectors \(a = a_1, \cdots, a_n\) such that \( R_a(x) =0 \) modulo \(G(x)\), where \( R_a(x) = \sum_{i=1}^n \frac{a_i}{z - \alpha_i} \).
- Gottesman-Kitaev-Preskill (GKP) code[452,453] Quantum lattice code for a non-degenerate lattice, thereby admitting a finite-dimensional logical subspace. Codes on \(n\) modes can be constructed from lattices with \(2n\)-dimensional full-rank Gram matrices \(A\).
- Graph quantum code[454] A stabilizer code on tensor products of \(G\)-valued qudits for Abelian \(G\) whose encoding isometry is defined using a graph [454; Eqs. (4-5)]. An analytical form of the codewords exists in terms of the adjacency matrix of the graph and bicharacters of the Abelian group [454]; see [455; Eq. (1)]. A graph quantum code for \(G=\mathbb{Z}_2\) contains a cluster state as one of its codewords and reduces to a cluster state when its logical dimension is one [456].
- Graph-adjacency code[457,458] Binary linear code whose generator matrix forms the adjacency matrix of a strongly regular graph. Given an adjacency matrix \(A\), the generator matrix is either \(G=A\) or \(G=(I|A)\), where \(I\) is the identity matrix.
- Grassmannian code[459–461] Evaluation code of polynomials evaluated on points lying on a Grassmannian \({\mathbb{G}}(\ell,m)\) [462].
- Gray code[463–465] The first Gray code [463], now called the binary reflected Gray code, is a trivial \([n,n,1]\) code that orders length-\(n\) binary strings such that nearest-neighbor strings differ by only one digit.
- Griesmer code[466–468] A type of \(q\)-ary code whose parameters satisfy the Griesmer bound with equality.
- Group GKP code[263] Group-based quantum code whose construction is based on nested subgroups \(H\subset K \subset G\). Logical subspace is spanned by basis states that are equal superpositions of elements of cosets of \(H\) in \(K\), and can be finite- or infinite-dimensional.
- Group-algebra code[469] a.k.a. \(G\) code.An \( [n,k]_q \) code whose automorphism group includes a finite group \( G \) of size \(n \), which acts on the code via its regular representation. This makes the code a \(G\)-submodule of the module \(GF(q)^n\) [471][470; Lemma 2.3]. A group-algebra code for an Abelian group is called an Abelian group-algebra code.
- Group-alphabet code Encodes \(K\) states (codewords) in coordinates labeled by elements of a group \(G\). The number of codewords may be infinite for infinite groups, so various restricted versions have to be constructed in practice.
- Group-based QPC[345] An \([[m r,1,\min(m,r)]]_G\) generalization of the QPC.
- Group-based cluster-state code[472] A code based on a group-based cluster state for a finite group \(G\) [472]. Such cluster states can be defined using a graph and conditional group multiplication operations. A group-based cluster state for \(G=GF(q)\) for prime-power \(q\) is called a Galois-qudit cluster state, while the state for \(G=\mathbb{Z}_q\) for positive \(q\) is called a modular-qudit cluster state.
- Group-based quantum code Encodes a logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space of \(L^2\)-normalizable functions on a second-countable unimodular group \(G\), i.e., a \(G\)-valued qudit. In other words, a group-valued qudit is a vector space whose canonical basis states \(|g\rangle\) are labeled by elements \(g\) of a group \(G\). For \(K\)-dimensional logical subspace and for block codes defined on groups \(G^{n}\), can be denoted as \(((n,K))_G\). When the logical subspace is the Hilbert space of \(L^2\)-normalizable functions on \(G^{ k}\), can be denoted as \([[n,k]]_G\). Ideal codewords may not be normalizable, depending on whether \(G\) is continuous and/or noncompact, so approximate versions have to be constructed in practice.
- Group-based quantum repetition code[345] An \([[n,1]]_G\) generalization of the quantum repetition code.
- Group-orbit code Code whose set of codewords forms an orbit of some reference codeword under a subgroup of the automorphism group, i.e., the group of distance-preserving transformations on the metric space defined with the code's alphabet.
- Group-representation code[175,176,473] Code whose projection is onto an irreducible representation of a subgroup \(G\) of a group of canonical or distinguished unitary operations, e.g., transversal gates in the case of block quantum codes, Gaussian operations in the case of bosonic codes, or \(SU(2)\) operations in the case of single-spin codes.
- Groupoid toric code[474] Extension of the Kitaev surface code from Abelian groups to groupoids, i.e., multi-fusion categories in which every morphism is an isomorphism [475]. Some models admit fracton-like features such as extensive ground-state degeneracy and excitations with restricted mobility. The robustness of these features has not yet been established.
- Guth-Lubotzky code[476] Hyperbolic surface code based on cellulations of certain four-dimensional manifolds. The manifolds are shown to have good homology and systolic properties for the purposes of code construction, with corresponding codes exhibiting linear rate.
- Haah cubic code (CC)[361] A 3D lattice stabilizer code on a length-\(L\) cubic lattice with one or two qubits per site. Admits two types of stabilizer generators with support on each cube of the lattice. In the non-CSS case, these two are related by spatial inversion. For CSS codes, we require that the product of all corner operators is the identity. We lastly require that there are no non-trival string operators, meaning that single-site operators are a phase, and any period one logical operator \(l \in \mathsf{S}^{\perp}\) is just a phase.
- Haar-random qubit code[477–480] Haar-random codewords are generated in a process involving averaging over unitary operations distributed accoding to the Haar measure. Haar-random codes are used to prove statements about the capacity of a quantum channel to transmit quantum information [481], but encoding and decoding in such \(n\)-qubit codes quickly becomes impractical as \(n\to\infty\).
- Hadamard BPSK c-q code[90] Multimode coherent-state c-q code that is a concatenation of a Hadamard code with a BPSK c-q code. Its codewords are \(n\)-mode coherent states whose components \(\pm\alpha\) are arranged according to rows of a Hadamard matrix.
- Hadamard code a.k.a. Walsh code, Walsh-Hadamard code.An \([2^m,m,2^{m-1}]\) balanced binary code. The \([2^m,m+1,2^{m-1}]\) augmented Hadamard code is the first-order RM code (a.k.a. RM\((1,m)\)), while the \([2^m-1,m,2^{m-1}]\) shortened Hadamard code is the simplex code (a.k.a. RM\(^*(1,m)\)).
- Hamiltonian-based code Code whose codespace corresponds to a set of energy eigenstates of a quantum-mechanical Hamiltonian i.e., a Hermitian operator whose expectation value measures the energy of its underlying physical system. The codespace is typically a set of low-energy eigenstates or ground states, but can include subspaces of arbitrarily high energy. Hamiltonians whose eigenstates are the canonical basis elements are called classical; otherwise, a Hamiltonian is called quantum.
- Hansen toric code[482,483] Evaluation code of a linear space of polynomials evaluated on points lying on an affine or projective toric variety. If the space is taken to be all polynomials up to some degree, the code is called a toric RM-type code of that degree.
- Hastings-Haah Floquet code[293] DA code whose sequence of check-operator measurements is periodic. The first instance of a dynamical code.
- Hayden-Nezami-Salton-Sanders bosonic code[484] An \([[n,1]]_{\mathbb{R}}\) analog CSS code defined using homological structres associated with an \(n-1\) simplex. Relevant to the study of spacetime replication of quantum information [485].
- Heavy-hexagon code[486] Subsystem stabilizer code on the heavy-hexagonal lattice that combines Bacon-Shor and surface-code stabilizers. Encodes one logical qubit into \(n=(5d^2-2d-1)/2\) physical qubits with distance \(d\). The heavy-hexagonal lattice allows for low degree (at most 3) connectivity between all the data and ancilla qubits, which is suitable for fixed-frequency transom qubits subject to frequency collision errors. The code can be split into a surface and a Bacon-Shor code, with the idling qubits of one code serving as the physical qubits of the other [487].
- Hemicubic code[488]
- Heptagon holographic code[489] a.k.a. Holographic Steane code.Holographic tensor-network code constructed out of a network of encoding isometries of the Steane code. Depending on how the isometry tensors are contracted, there is a zero-rate and a finite-rate code family.
- Hergert code[256] a.k.a. Goethals-Delsarte (GD) code.A nonlinear subcode of an RM code that is a formal dual of the nonlinear DG code in the sense that its distance distribution is equal to the MacWilliams transform of the distance distribution of a DG codes.
- Hermitian Galois-qudit code[415,416,490] a.k.a. \(GF(q^2)\)-linear code.An \([[n,k,d]]_q\) true Galois-qudit stabilizer code constructed from a Hermitian self-orthogonal linear code over \(GF(q^2)\) using the one-to-one correspondence between the Galois-qudit Pauli matrices and elements of the Galois field \(GF(q^2)\).
- Hermitian code[491,492][325; Sec. 4.4.3] Evaluation AG code of rational functions evaluated on points lying on a Hermitian curve in either affine or projective space. Hermitian codes improve over RS codes in length: that RS codes have length at most \(q+1\) while Hermitian codes have length \(q^3 + 1\).
- Hermitian qubit code[282] a.k.a. Calderbank-Rains-Shor-Sloane (CRSS) code, \(GF(4)\)-linear code.An \([[n,k,d]]\) stabilizer code constructed from a Hermitian self-orthogonal linear quaternary code using the \(GF(4)\) representation.
- Hermitian-hypersurface code[493] Evaluation code of polynomials evaluated on points lying on a Hermitian hypersurface.
- Hessian QSC[178]
Quantum spherical code encoding a logical qubit, with each codeword an equal superposition of vertices of a Hessian complex polyhedron. For the unit sphere, the codewords are \begin{align} |\overline{0}\rangle &= \frac{1}{\sqrt{27}}\left( \sum_{\mu,\nu=0}^{2} |0,\omega^{\mu},-\omega^{\nu}\rangle + |-\omega^{\nu},0,\omega^{\mu}\rangle + |\omega^{\mu},-\omega^{\nu},0\rangle \right) \tag*{(6)}\\ |\overline{1}\rangle &= \frac{1}{\sqrt{27}}\left( \sum_{\mu,\nu=0}^{2} |0,-\omega^{\mu},\omega^{\nu}\rangle + |\omega^{\nu},0,-\omega^{\mu}\rangle + |-\omega^{\mu},\omega^{\nu},0\rangle \right)~, \tag*{(7)}\end{align} where \(\omega = e^{\frac{2\pi i}{3}}\).

- Hessian polyhedron code[494,495] a.k.a. Schläfli configuration.Spherical \((6,27,3/2)\) code whose codewords are the vertices of the Hessian complex polyhedron and the \(2_{21}\) real polytope. Two copies of the code yield the \((6,54,1)\) double Hessian polyhedron (a.k.a. diplo-Schläfli) code. The code can be obtained from the Schläfli graph [116; Ch. 9]. The (antipodal pairs of) points of the (double) Hessian polyhedron correspond to the 27 lines on a smooth cubic surface in the complex projective plane [150,496–498].
- Hexacode[110,499] The \([6,3,4]_4\) self-dual MDS code that has connections to projective geometry, lattices [110], and conformal field theory [500]. Puncturing the code yields the perfect \([5,3,3]_4\) quaternary Hamming code known as the shortened hexacode or shorter hexacode [501]. Both codes are sometimes refereed to as Golay codes over \(GF(4)\).
- Hexagonal GKP code[452] Single-mode GKP qudit-into-oscillator code based on the hexagonal lattice. Offers the best error correction against displacement noise in a single mode due to the optimal packing of the underlying lattice.
- Hierarchical code[502] Member of a family of \([[n,k,d]]\) qubit stabilizer codes resulting from a concatenation of a constant-rate QLDPC code with a rotated surface code. Concatenation allows for syndrome extraction to be performed on a 2D geometry while maintining a threshold at the expense of a logarithmically vanishing rate. The growing syndrome extraction circuit depth allows known bounds in the literature to be weakened [503,504].
- High-dimensional expander (HDX) code[279,505] CSS code constructed from a Ramanujan quantum code and an asymptotically good classical LDPC code using distance balancing. Ramanujan quantum codes are defined using Ramanujan complexes which are simplicial complexes that generalise Ramanujan graphs [330,331]. Combining the quantum code obtained from a Ramanujan complex and a good classical LDPC code, which can be thought of as coming from a 1-dimensional chain complex, yields a new quantum code that is defined on a 2-dimensional chain complex. This 2-dimensional chain complex is obtained by the co-complex of the product of the 2 co-complexes. The length, dimension and distance of the new quantum code depend on the input codes.
- Higman-Sims graph-adjacency code[457,458] A graph-based code whose generator matrix is constructed using the adjacency matrix \(A\) of the Higman-Sims graph. Setting the generator matrix \(G=(I|A)\) yields a \([100,22,32]\) code whose dual is an optimal \([100,78,8]\) code [457; Table VI].
- Hill projective-cap code[506] Member of a projective code family that contains of \(q\)-ary sharp configurations and that is constructed using projective caps.
- Hirschfeld code[507] A projective geometry code that is an example of an MDS code that is not an RS code; see [508; Exam. 7.6] for the description.
- Hoffman-Singleton cycle code[457,458] A \([50,21,12]\) cycle code whose parity-check matrix is the incidence matrix of the Hoffman-Singleton graph [509]. Its dual is a \([50,29,8]\) code [457; Table II].
- Hoffman-Singleton graph-adjacency code[457,458] A graph-based code whose generator matrix is constructed using the adjacency matrix of the Hoffman-Singleton graph [509]. Setting the generator matrix equal to the adjacency matrix, \(G=A\), yields a \([50,22,7]\) code whose dual is a \([50,28,8]\) code [457; Table III].
- Holographic code[510] Block quantum code whose features serve to model aspects of the AdS/CFT holographic duality and, more generally, quantum gravity.
- Holographic hybrid code[511] a.k.a. Subsystem holographic code.Holographic tensor-network code constructed out of alternating isometries of the five-qubit and \([[4,1,1,2]]\) Bacon-Shor codes.
- Holographic tensor-network code[510,512–514] Quantum Lego code whose encoding isometry forms a holographic tensor network, i.e., a tensor network associated with a tiling of hyperbolic space. Physical qubits are associated with uncontracted tensor legs at the boundary of the tesselation, while logical qubits are associated with uncontracted legs in the bulk. The number of layers emanating form the central point of the tiling is the radius of the code.
- Homological code[24,365,515,516] a.k.a. Generalized surface code.CSS-type extenstion of the Kitaev surface code to arbitrary manifolds. The version on a Euclidean manifold of some fixed dimension is called the \(D\)-dimensional "surface" or \(D\)-dimensional toric code.
- Homological number-phase code[517] A homological \(n\)-rotor code mapped into the Fock-state space of \(n\) oscillators by identifying non-negative rotor angular-momentum states with oscillator Fock states. The resulting oscillator code can encode logical rotors or qudits due to the presence of torsion in the chain complex defining the original rotor code.
- Homological product code[518,519] a.k.a. Tensor product code.CSS code formulated using the tensor product of two chain complexes (see Qubit CSS-to-homology correspondence).
- Homological rotor code[520] A homological quantum rotor code is an extension of analog stabilizer codes to rotors. The code is stabilized by a continuous group of rotor \(X\)-type and \(Z\)-type generalized Pauli operators. Codes are formulated using an extension of the qubit CSS-to-homology correspondence to rotors. The homology group of the logical operators has a torsion component because the chain complexes are defined over the ring of integers, which yields codes with finite logical dimension, i.e., encoding logical qudits instead of only logical rotors. Such finite-dimensional encodings are not possible with analog stabilizer codes.
- Honeycomb (6.6.6) color code[7] Triangular color code defined on a patch of the 6.6.6 (honeycomb) tiling.
- Honeycomb Floquet code[293] Floquet code based on the Kitaev honeycomb model [521] whose logical qubits are generated through a particular sequence of measurements. A CSS version of the code has been proposed which loosens the restriction of which sequences to use [348]. The code has also been generalized to arbitrary non-chiral, Abelian topological order [522].
- Hopf-algebra cluster-state code[523] Code based on a cluster state defined on qudits valued in a Hopf algebra.
- Hopf-algebra quantum-double code[524,525] Code whose codewords realize 2D gapped topological order defined on qudits valued in a Hopf algebra \(H\). The code Hamiltonian is an generalization [524,525] of the quantum double model from group algebras to Hopf algebras, as anticipated by Kitaev [35]. Boundaries of these models have been examined [526,527].
- Hsieh-Halasz (HH) code[528] Member of one of two families of fracton codes, named HH-I and HH-II, defined on a cubic lattice with two qubits per site. HH-I (HH-II) is a CSS (non-CSS) stabilizer code family, with the former identified as a foliated type-I fracton code [19].
- Hsieh-Halasz-Balents (HHB) code[529] Member of one of two families of fracton codes, named HHB model A and B, defined on a cubic lattice with two qubits per site. Both are expected to be foliated type-I fracton codes [19; Eqs. (D42-D43)].
- Hsu-Anastasopoulos LDPC (HA-LDPC) code[530] A regular LDPC code obtained from a concatenation of a certain random regular LDPC code and a certain random LDGM code. Using rate-one LDGM codes eliminates high-weight codewords while admitting an amount of low-weight codewords that asymptotically vanishes, allowing code families to achieve the GV bound with high probability.
- Hybrid QECC[315,531–533] A quantum code which encodes both quantum and classical information.
- Hybrid cat code[534,535] A hybrid qubit-oscillator code admitting codewords that are tensor products of a single-qubit (e.g., photon polarization) state with either a cat state or a coherent state.
- Hybrid convolutional code[536] A quantum convolutional code which protects both quantum and classical information.
- Hybrid qubit code[315,537] A qubit code which stores both quantum and classical information. Usually denoted as \(((n,K:M))\) or \(((n,K:M,d))\), where \(K\) is the dimension of the underlying quantum code, where \(M\) is the size of the classical code, and where \(d\) is the distance.
- Hybrid qudit-oscillator code Encodes a \(K\)-dimensional logical Hilbert space into \(n_1\) modular qudits of dimension \(q\) and \(n_2 \neq 0\) oscillators, i.e., the Hilbert space of \(L^2\)-normalizable functions on \(\mathbb{Z}_q^{n_1} \times \mathbb{R}^{n_2}\). In photonic systems, photonic states of multiple degrees of freedom of a photon (e.g., frequency, amplitude, and polarization) are called hyper-entangled states [538].
- Hybrid stabilizer code[315,537] A qubit stabilizer code which stores both quantum and classical information. Usually denoted as \([[n,k:c]]\) or \([[n,k:c,d]]\), where \(k\) (\(c\)) is the number of encoded qubits (classical bits), and where \(d\) is the distance.
- Hyperbolic Floquet code[539–541] Floquet code whose check-operators correspond to edges of a hyperbolic lattice of degree 3.
- Hyperbolic color code[542–544] An extension of the color code construction to hyperbolic manifolds. As opposed to there being only three types of uniform three-valent and three-colorable lattice tilings in the 2D Euclidean plane, there is an infinite number of admissible hyperbolic tilings in the 2D hyperbolic plane [543]. Certain double covers of hyperbolic tilings also yield admissible tilings [542]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [8]; see also a construction based on the more general quantum pin codes [544].
- Hyperbolic evaluation code[545–547] a.k.a. Hyperbolic cascaded RS code.An evaluation code over polynomials in two variables. Generator matrices are determined in Ref. [547] following initial formulations of the codes as generalized concatenations of RS codes [545,546]; see [324; Ex. 4.26].
- Hyperbolic sphere packing[548,549] Encodes states (codewords) into coordinates in the hyperbolic plane \(\mathbb{H}^2\).
- Hyperbolic surface code An extension of the Kitaev surface code construction to hyperbolic manifolds. Given a cellulation of a manifold, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \(i+1\)-faces.
- Hypercube code Spherical \((n,2^n,4/n)\) code whose codewords are vertices of an \(n\)-cube, i.e., all permutations and negations of the vector \((1,1,\cdots,1)\), up to normalization.
- Hypergraph product (HGP) code[550,551] a.k.a. Quantum hypergraph (QHG) code, Tillich-Zemor product code.A member of a family of CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear codes. Codes from hypergraph products in higher dimension are called higher-dimensional HGP codes [552].
- Hyperinvariant tensor-network (HTN) code[553] a.k.a. Evenbly code.Holographic tensor-network error-detecting code constructed out of a hyperinvariant tensor network [554], i.e., a MERA-like network admitting a hyperbolic geometry. The network is defined using two layers A and B, with constituent tensors satisfying isometry conditions (a.k.a. multitensor constraints).
- Hyperoval code[555] A projective code constructed using hyperovals in projective space.
- Hypersphere product code[556]
- Icosahedron code Spherical \((3,12,2-2/\sqrt{5})\) code whose codewords are the vertices of the icosahedron (alternatively, the centers of the faces of a dodecahedron, the icosahedron's dual polytope).
- Identifiable parent property (IPP) code[557] A code that is embedded in copyrighted content in order to detect unauthorized redistribution of said content by pirates. IPP codes are designed to detect pirates even when segments content are mixed together so as to conceal the pirates' identities.
- Incidence-matrix projective code[558–560] Code whose generator matrix is the incidence matrix of points and hyperplanes in a projective space. Has been generalized to incidence matrices of other structures [561,562][563; Sec. 14.4]. Columns of a code's parity-check matrix can similarly correspond to an incidence matrix.
- Interleaved RS (IRS) code A modification of RS codes where multiple polynomials are used to define each codeword. Each codeword \(\mu\) of a \(t\)-interleaved RS code is a string of values of the corresponding set \(\{f_\mu^{(1)},f_\mu^{(2)},\cdots,f_\mu^{(t)}\}\) of \(t\) polynomials at the points \(\alpha_i\). The vector codewords can be arranged in an array whose rows are ordinary RS codes for each polynomial \(f^{j}\), yielding the encoding \begin{align} \mu\to\left( \begin{array}{cccc} f_{\mu}^{(1)}\left(\alpha_{1}\right) & f_{\mu}^{(1)}\left(\alpha_{2}\right) & \cdots & f_{\mu}^{(1)}\left(\alpha_{n}\right)\\ f_{\mu}^{(2)}\left(\alpha_{1}\right) & f_{\mu}^{(2)}\left(\alpha_{2}\right) & & f_{\mu}^{(2)}\left(\alpha_{n}\right)\\ \vdots & & \ddots & \vdots\\ f_{\mu}^{(t)}\left(\alpha_{1}\right) & f_{\mu}^{(t)}\left(\alpha_{2}\right) & \cdots & f_{\mu}^{(t)}\left(\alpha_{n}\right) \end{array}\right)~. \tag*{(8)}\end{align}
- Irregular LDPC code[564,565] An LDPC code whose parity-check matrix has a variable number of entries in each row or column.
- Irregular repeat-accumulate (IRA) code[566–568] A generalization of the RA code in which the outer 1-in-3 repetition encoding step is replaced by an LDGM code. A simple version is when different bits in the RA block are repeated a different number of times.
- Jordan-Wigner transformation code[569–571] A mapping between qubit Pauli strings and Majorana operators that can be thought of as a trivial \([[n,n,1]]\) code. The mapping is best described as converting a chain of \(n\) qubits into a chain of \(2n\) Majorana modes (i.e., \(n\) fermionic modes). It maps Majorana operators into Pauli strings of weight \(O(n)\).
- Julin-Golay code[109,572,573] One of several nonlinear binary \((12,144,4)\) codes based on the Steiner system \(S(5,6,12)\) [574,575][56; Sec. 2.7][576; Sec. 4] or their shortened versions, the nonlinear \((11,72,4)\), \((10,38,4)\), and \((9,20,4)\) Julin-Golay codes. Several of these codes contain more codewords than linear codes of the same length and distance and yield non-lattice sphere-packings that hold records in 9 and 11 dimensions.
- Jump code[577–579] A CE code designed to detect and correct AD errors. An \(((n,K))\) jump code is denoted as \(((n,K,t))_w\) (which conflicts with modular-qudit notation), where \(t\) is the maximum number of qubits that can be corrected after each one has undergone a jump error \(|0\rangle\langle 1|\), and where each codeword is a uniform superposition of qubit basis states with Hamming weight \(w\).
- Justesen code[580] Binary linear code resulting from generalized concatenation of an outer RS code with multiple inner codes sampled from the Wozencraft ensemble, i.e., \(N\) distinct binary inner codes of dimension \(m\) and length \(2m\). The first asymptotically good codes.
- Kasami code[581] Member of the family of \([2^{2r}-1, 3r, 2^{2r-1} - 2^{r-1} ]\) cyclic binary linear codes.
- Kerdock code[582] Binary nonlinear \((2^m, 2^{2m}, 2^{m-1} - 2^{(m-2)/2})\) for even \(m\) consisting of the first-order Reed-Muller code RM\((1,m)\) with maximum-rank cosets of RM\((1,m)\) in RM\((2,m)\).
- Kerdock spherical code[583–585] Family of \((n=2^{2r},n^2,2-2/\sqrt{n})\) spherical codes for \(r \geq 2\), obtained from Kerdock codes via the antipodal mapping [116; pg. 157]. These codes are optimal for their parameters for \(2\leq r\leq 5\), they are unique for \(r\in\{2,3\}\), and they form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) cross polytopes [586].
- Kim-Preskill-Tang (KPT) code[587] A quantum error-correcting code that protects the encoded interior of a black hole from computationally bounded exterior observers. Under the assumption that the Hawking radiation emitted by an old black hole is pseudorandom, there exists a subspace of the radiation system that encodes the black hole interior, entangled with the late outgoing Hawking quanta. The logical operators of this code commute with efficient operations acting on the radiation, protecting the interior up to corrections exponentially small in the black hole's entropy.
- Kitaev chain code[570] An \([[n,1,1]]_{f}\) Majorana stabilizer code forming the ground-state of the Kitaev Majorana chain (a.k.a. Kitaev Majorana wire) in its fermionic topological phase, which is unitarily equivalent to the 1D quantum Ising model in the symmetry-breaking phase via the Jordan-Wigner transformation. The code is usually defined using the algebra of two anti-commuting Majorana operators called Majorana zero modes (MZMs) or Majorana edge modes (MEMs).
- Kitaev current-mirror qubit code[520,588,589]
- Kitaev honeycomb code[434,521,591] Code whose logical subspace is labeled by different fusion outcomes of Ising anyons present in the Ising-anyon topological phase of the Kitaev honeycomb model [521]. Each logical qubit is constructed out of four Majorana operators, which admit braiding-based gates due to their non-Abelian statistics and which can be used for topological quantum computation. Ising anyons also exist in other phases, such as the fractional quantum Hall phase [592].
- Kitaev surface code[35,75,593,594] A family of Abelian topological CSS stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a cellulation of a two-dimensional surface (with a qubit located at each edge of the cellulation). Codewords correspond to ground states of the surface code Hamiltonian, and error operators create or annihilate pairs of anyonic charges or vortices.
- Klein-quartic code[595] Evaluation AG code over \(GF(8)\) of rational functions evaluated on points lying on the Klein quartic, which is defined by the equation \(x^3 y + y^3 z + z^3 x = 0\) ([324], Ex. 2.75).
- Knill code[398] a.k.a. Clifford code.A group representation code whose projection is onto an irrep of a normal subgroup of the group formed by a nice error basis. Knill codes yield stabilizer-like codes based on error bases that are non-Pauli but that nevertheless maintain many of the useful features of Pauli-type bases.
- Kopparty-Meir-Ron-Zewi-Saraf (KMRS) code[596,597] Member of a family of locally testable binary linear codes with constant rate, constant relative distance, and subpolynomial query complexity \(u = (\log n)^{O(\log \log n)}\)). Later work by Gopi, Kopparty, Oliveira, Ron-Zewi, and Saraf [597] showed that related concatenated codes achieve the GV bound.
- LDPC convolutional code (LDPC-CC)[598–600] a.k.a. Low-density convolutional (LDC) code.Convolutional code defined by an infinite low-density parity-check matrix.
- La-cross code[601] Code constructed using the hypergraph product of two copies of a cyclic LDPC code. The construction uses cyclic LDPC codes with generating polynomials \(1+x+x^k\) for some \(k\). Using a length-\(n\) seed code yields an \([[2n^2,2k^2]]\) family for periodic boundary conditions and an \([[(n-k)^2+n^2,k^2]]\) family for open boundary conditions.
- Ladder Floquet code[293] Floquet code defined on a ladder qubit geometry, with one qubit per vertex. The check operators consist of \(ZZ\) checks on each rung and alternating \(XX\) and \(YY\) check on the legs.
- Laminated spherical code[602] Spherical code whose codewords are obtained from a recursive procedure that is similar to the procedure that creates laminated lattices.
- Landau-level spin code[603] Approximate quantum code that encodes a qudit in the finite-dimensional Hilbert space of a single spin, i.e., a spherical Landau level. Codewords are approximately orthogonal Landau-level spin coherent states whose orientations are spaced maximally far apart along a great circle (equator) of the sphere. The larger the spin, the better the performance.
- Lattice stabilizer code[17,361,604] a.k.a. Topological stabilizer code.A geometrically local modular-qudit or Galois-qudit stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\). On an infinite lattice, its stabilizer group is generated by few-site Pauli operators and their translations, in which case the code is called translationally invariant stabilizer code. Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional versions. Lattice defects and boundaries between different codes can also be introduced.
- Lattice subsystem code[12] a.k.a. Topological subsystem code.A geometrically local qubit, modular-qudit, or Galois-qudit subsystem stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\). On an infinite lattice, its gauge and stabilizer groups are generated by few-site Pauli operators and their translations, in which case the code is called translationally invariant subsystem code. Boundary conditions have to be imposed on the lattice in order to obtain finite-dimensional versions, in which case the stabilizer group may no longer be generated by few-site Pauli operators. Lattice defects and boundaries between different codes can also be introduced.
- Lattice-based code Encodes states (codewords) in coordinates of an \(n\)-dimensional lattice, i.e., a discrete set of points in Euclidean space \(\mathbb{R}^n\) that forms a group under vector addition when the set is translated such that one point is at the origin. The number of codewords may be infinite because the coordinate space is infinite-dimensional, so various restricted versions have to be constructed in practice. Since lattices are closed under addition, lattice-based codes can be thought of as linear codes over the reals.
- Lattice-shell code[605,606] Spherical code whose codewords are scaled versions of points on a lattice. A \(m\)-shell code consists of normalized lattice vectors \(x\) with squared norm \(\|x\|^2 = m\). Each code is constructed by normalizing a set of lattice vectors in one or more shells, i.e., sets of lattice points lying on a hypersphere.
- Layer code[607] Member of a family of 3D lattice CSS codes with stabilizer generator weights \(\leq 6\) that are obtained by coupling layers of 2D surface code according to the Tanner graph of a QLDPC code. Geometric locality is maintained because, instead of being concatenated, each pair of parallel surface-code squares are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery.
- Lazebnik-Ustimenko (LU) code[608,609] LDPC code whose Tanner graph comes from a particular family of \(q\)-regular graphs [608] of known girth and relatively large stopping sets.
- Lechner-Hauke-Zoller (LHZ) code[610,611] a.k.a. Lechner-Hauke-Zoller (LHZ) parity code.LDPC c-q code designed to convert the long-range interactions of a quantum annealer into local constraints. The code maps the bits of a classical Ising model with all-to-all \(D\)-body interactions into one on a \(D\)-dimensional lattice. An extension maps more general models onto the same lattice [612].
- Left-right Cayley complex code[613] Binary code constructed on a left-right Cayley complex using a pair of base codes \(C_A,C_B\) and an expander graph [328] such that codewords for a fixed graph vertex are codewords of the tensor code \(C_A \otimes C_B\). A family of such codes is one of the first \(c^3\)-LTCs.
- Levenshtein code[614] Binary codes constructed from combining two codes \(A'\) constructed out of Hadamard matrices.
- Lexicographic code[615,616] A \(q\)-ary code whose codewords are constructed greedily and iteratively by starting with zero and adding codewords whose distance is the desired minimum distance of the code.
- Lift-connected surface (LCS) code[617] Member of one of several families of lifted-product codes that consist of sparsely interconnected stacks of surface codes.
- Lifted-product (LP) code[28,327] a.k.a. Panteleev-Kalachev (PK) code.Code that utilizes the notion of a lifted product in its construction. Lifted products of certain classical Tanner codes are the first (asymptotically) good QLDPC codes.
- Linear STC Spacetime code whose set of matrix codewords is closed under addition and subtraction.
- Linear \(q\)-ary code An \((n,K,d)_q\) linear code is denoted as \([n,k,d]_q\), where \(k=\log_q K\) need not be an integer. Its codewords form a linear subspace, i.e., for any codewords \(x,y\), \(\alpha x+ \beta y\) is also a codeword for any \(q\)-ary digits \(\alpha,\beta\). This extra structure yields much information about their properties, making them a large and well-studied subset of codes.
- Linear binary code An \((n,2^k,d)\) linear code is denoted as \([n,k]\) or \([n,k,d]\), where \(d\) is the code's distance. Its codewords form a linear subspace, i.e., for any codewords \(x,y\), \(x+y\) is also a codeword. A code that is not linear is called nonlinear.
- Linear code over \(G\)[618–620] Block code that encodes \(K\) states (codewords) in \(n\) coordinates over a group \(G\) such that the codewords form a subgroup of \(G^n\). In other words, the set of codewords is closed under the group operation.
- Linear code with complementary dual (LCD)[621] A linear code \(C\) admits a complementary dual if \(C\) and its dual code \(C^{\perp}\) do not share any codewords.
- Linearized RS code[622–624]
- Local Haar-random circuit qubit code[625] An \(n\)-qubit code whose codewords are a pair of approximately locally indistinguishable states produced by starting with any two orthogonal \(n\)-qubit states and acting with a random unitary circuit of depth polynomial in \(n\). Two states are locally indistinguishable if they cannot be distinguished by local measurements. A single layer of the encoding circuit is composed of about \(n/2\) two-qubit nearest-neighbor gates run in parallel, with each gate drawn randomly from the Haar distribution on two-qubit unitaries.
- Locally correctable code (LCC) Recall that a block code encodes a length-\(k\) message into a length-\(n\) codeword, which is then sent through a noise channel to yield an error word. Informally, an LCC is a block code for which one can recover any coordinate of a codeword from at most \(r\) coordinates of the error word (assuming the error word is within some tolerated corruption rate \(\delta\)).
- Locally decodable code (LDC)[626] Recall that a block code encodes a length-\(k\) message into a length-\(n\) codeword, which is then sent through a noise channel to yield an error word. Informally, an LDC is a block code for which one can recover any coordinate of the message from at most \(r\) coordinates of the error word (assuming the error word is within some tolerated corruption rate \(\delta\)). Efficiency of the correction is quantified by the code's query complexity \(r\), and correction is performed by sampling subsets of \(r\) bits.
- Locally recoverable code (LRC) a.k.a. Locally repairable code.An LRC is a block code for which one can recover any coordinate of a codeword from at most \(r\) other coordinates of the codeword. In other words, an LRC of locality \(r\) is a block code for which, given a codeword \(c\) and coordinate \(c_i\), \(c_i\) can be recovered from at most \(r\) other coordinates of \(c\). An \(r\)-locally recoverable code of length \(n\) and dimension \(k\) is denoted as an \((n,k,r)\) LRC. The definition can be generalized to \(t\)-LRC, if every coordinate is recoverable from \(t\) disjoint subsets of coordinates.
- Locally testable code (LTC)[627–630] Code for which one can efficiently check whether a given string is a codeword or is far from a codeword. Efficiency of the verification is quantified by the code's query complexity \(u\), while effectiveness is quantified by the code's soundness \(R\).
- Long code[631,632] Locally testable \([2^{2^k},k,d]\) code. The encoder maps a \(k\)-bit string into a codeword that consists of the values of all Boolean functions on the \(k\)-bit string. The code is not practical, but is useful for certain probabilistically checkable proof (PCP) constructions [633].
- Long-range enhanced surface code (LRESC)[634] Code constructed using the hypergraph product of two copies of a concatenated LDPC-repetition seed code. This family interpolates between surface codes and hypergraph codes since the hypergraph product of two repetition codes yields the planar surface code. The construction uses small \([3,2,2]\) and \([6,2,4]\) LDPC codes concatenated with \([4,1,4]\) and \([2,1,2]\) repetition codes, respectively. An example using a \([5,2,3]\) code is also presented.
- Loop toric code[24] a.k.a. Kitaev tesseract code, 4D surface code, All-loop toric code, \((2,2)\) 4D toric code.A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((2,2)\) toric code because it admits 2D membrane \(Z\)-type and \(X\)-type logical operators. Both types of operators create 1D (i.e., loop) excitations at their edges. The code serves as a self-correcting quantum memory [24,635].
- Lossless expander balanced-product code[636,637] QLDPC code constructed by taking the balanced product of lossless expander graphs. Using one part of a quantum-code chain complex constructed with one-sided loss expanders [638] yields a \(c^3\)-LTC [636]. Using two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [637].
- Low-density generator-matrix (LDGM) code Binary linear code with a sparse generator matrix. Alternatively, a member of an infinite family of \([n,k,d]\) codes for which the number of nonzero entries in each row and column of the generator matrix are both bounded by a constant as \(n\to\infty\). The dual of an LDGM code has a sparse parity-check matrix and is called an LDPC code.
- Low-density parity-check (LDPC) code[385,386] a.k.a. Sparse graph code.A binary linear code with a sparse parity-check matrix. Alternatively, a member of an infinite family of \([n,k,d]\) codes for which the number of nonzero entries in each row and column of the parity-check matrix are both bounded above by a constant as \(n\to\infty\).
- Low-rank parity-check (LRPC) code[639] An LRPC code of rank \(d\) is a rank-metric code that, when interpreted as a linear code over \(GF(q^m)\), admits an \((n-k)\times n\) parity-check matrix whose entries span a subspace of \(GF(q^m)\) that is at most \(d\)-dimensional.
- Luby transform (LT) code[640] Erasure codes based on fountain codes. They improve on random linear fountain codes by having a much more efficient encoding and decoding algorithm.
- MDS array code An \((n,k,m)\) array code whose codewords can be recovered by any \(k\) out of \(n\) nodes, where each node stores a length-\(m\) column of the codeword. MDS array codes are MDS codes when each matrix codeword is treated as a vector by converting each column into a single coordinate via subpacketization.
- MacKay-Neal LDPC (MN-LDPC) code[641,642] Codes whose parity-check matrix is constructed non-deterministically via the MacKay-Neal prescription. The parity-check matrix of an \((l,r,g\))-MN-LDPC code is of the form \((H_1~H_2)\), where \(H_1\) is a random binary matrix of column weight \(l\) and row weight \(r\), and \(H_2\) is a random binary matrix of column and row weight \(g\) [643].
- Magnon code[644] An \(n\)-spin approximate code whose codespace of \(k=\Omega(\log n)\) qubits is efficiently described in terms of particular matrix product states or Bethe ansatz tensor networks. Magnon codewords are low-energy excited states of the frustration-free Heisenberg-XXX model Hamiltonian [644].
- Majorana box qubit[570,645,646] An \([[n,1,2]]_{f}\) Majorana stabilizer code forming the even-fermion-parity ground-state subspace of two parallel Kitaev Majorana chains in their fermionic topological phase. The \([[2,1,2]]_{f}\) version is called the tetron Majorana code. An \([[3,2,2]]_{f}\) extension using three Kitaev chains and housing two logical qubits of the same parity is called the hexon Majorana code. Similarly, octon, decon, and dodecon are codes defined by the positive-parity subspace of \(4\), \(5\), and \(6\) fermionic modes, respectively [647].
- Majorana checkerboard code[157] a.k.a. Majorana cubic model code.A Majorana analogue of the X-cube model defined on a cubic lattice. The code admits weight-eight Majorana stabilizer generators on the eight vertices of each cube of a checkerboard sublattice.
- Majorana color code[647–649] Majorana analogue of the color code defined on a 2D tricolorable lattice and constructed out of Majorana box qubit codes placed on patches of the lattice.
- Majorana loop stabilizer code (MLSC)[650] An single error-correcting fermion-into-qubit encoding defined on a 2D qubit lattice whose stabilizers are associated with loops in the lattice.
- Majorana stabilizer code[648] A stabilizer code whose stabilizers are products of an even number of Majorana fermion operators, analogous to Pauli strings for a traditional stabilizer code and referred to as Majorana stabilizers. The codespace is the mutual \(+1\) eigenspace of all Majorana stabilizers. In such systems, Majorana fermions may either be considered individually or paired into creation and annihilation operators for fermionic modes. Codes can be denoted as \([[n,k,d]]_{f}\) [651], where \(n\) is the number of fermionic modes (equivalently, \(2n\) Majorana modes).
- Majorana subsystem stabilizer code[652] A Majorana stabilizer code with some of its logical qubits denoted as gauge qubits and not used for storage of logical information.
- Majorana surface code[653,654] Majorana analogue of the surface code defined on a 2D lattice and constructed out of Majorana box qubit codes placed on patches of the lattice.
- Margulis LDPC code[39]
- Matching code[656] Member of a class of qubit stabilizer codes based on the Abelian phase of the Kitaev honeycomb model.
- Matrix-based code a.k.a. Two-dimensional code.Encodes \(K\) states (codewords) in an \(m\times n\)-dimensional matrix of coordinates over a field (e.g., the Galois field \(GF(q)\) or the complex numbers \(\mathbb{C}\)).
- Matrix-model code[657,658] Multimode-mode Fock-state bosonic approximate code derived from a matrix model, i.e., a non-Abelian bosonic gauge theory with a large gauge group. The model's degrees of freedom are matrix-valued bosons \(a\), each consisting of \(N^2\) harmonic oscillator modes and subject to an \(SU(N)\) gauge symmetry.
- Matrix-product code[659] Code constructed using a concatenation procedure that yields a code consisting of all products of codewords in \(M\) length-\(n\) \(q\)-ary codes and an \(M\times N\) \(q\)-ary matrix with \(N\geq M\). A prominent subclass is the case with \(A\) is non-singular by columns (NSC).
- Maximal-entanglement EA Galois-qudit stabilizer code[660,661] An \([[n,k,d;e]]_q\) EA Galois-qudit stabilizer code for which \(e = n-k\).
- Maximally recoverable (MR) code[662,663] a.k.a. Partial MDS code.A code with \((r,\delta)\) locality such that puncturing it on any \(\delta-1\) coordinates of the local \([r+\delta-1,r,\delta]\) codes yields an MDS code.
- Maximum distance separable (MDS) code[664] A type of \(q\)-ary code whose parameters satisfy the Singleton bound with equality.
- Maximum-rank distance (MRD) code[383,384,665] a.k.a. Optimal rank-distance code.An \([n\times m,k,d]_q\) rank-metric code whose parameters are such that the Singleton-like bound \begin{align} k \leq \max(n, m) (\min(n, m) - d + 1) \tag*{(9)}\end{align} becomes an equality.
- Maximum-sum-rank distance (MSRD) code[624] a.k.a. Optimal sum-rank-distance code.An \([n\times m,k,d]_q\) rank-metric code whose parameters are such that the sum-rank-metric Singleton bound [624; Prop. 34] \begin{align} d_{\text{SR}}(C) \leq n - k + 1 \tag*{(10)}\end{align} becomes an equality, where \(d_{\text{SR}}\) is the sum-rank metric.
- Meir code[666] Locally testable \([n,k,d]_q\) code with query complexity \(\text{poly}(\log k)\) and rejection ratio \(R/n = 1/\text{poly}(\log k)\). Code construction is probabilistic and combinatorial.
- Melas code[667,668] Cyclic \([2^m -1, 2^m - 1 - 2m, 5]\) linear code with generator polynomial is \(g(x) = p(x)p(x)^{\star}\), where \(p(x)\) is a primitive polynomial of degree \(m\) that is the minimal polynomial over \(GF(2)\) of an element \(\alpha\) of order \(2^m -1\) in \(GF(2^m)\), \(m\) is odd and greater that five, and '\(\star\)' denotes reciprocation [669].
- Metrological code[670] Two-dimensional subspace of a Hilbert space whose basis states satisfy only a part of the Knill-Laflamme conditions. The satisfied part of the conditions ensures that the code can be used for local parameter estimation.
- Minimum-bandwidth regenerating (MBR) code An RGC that corresponds to an extreme point in the storage-bandwidth trade-off curve that is characterised by \(\alpha = d\beta\).
- Minimum-storage regenerating (MSR) code An RGC that corresponds to an extreme point in the storage-bandwidth trade-off curve that is characterised by \(\alpha = (d-k+1)\beta\).
- Mixed code a.k.a. Mixed-alphabet code.Encodes \(K\) states (codewords) in a string of two or more coordinates, each of which takes values in one of two or more possible groups.
- Modular-qudit CSS code[395–397] An \(((n,K,d))_q\) modular-qudit stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. Codes can be defined from two classical codes and/or chain complexes over the ring \(\mathbb{Z}_q\) via an extension of qubit CSS-to-homology correspondence to modular qudits. The homology group of the logical operators has a torsion component because the chain complexes are defined over a ring, which yields codes whose logical dimension is not a power of \(q\).
- Modular-qudit CWS code[671–673] A CWS code for modular qudits, defined using a modular-qudit cluster state and a set of modular-qudit \(Z\)-type Pauli strings defined by a \(q\)-ary classical code over \(\mathbb{Z}_q\).
- Modular-qudit GKP code[452; Sec. II] a.k.a. Pre-GKP code.Modular-qudit analogue of the GKP code. Encodes a qudit into a larger qudit and protects against Pauli shifts up to some maximum value.
- Modular-qudit USt code[671,672] A modular-qubit code whose codespace consists of a direct sum of a modular-qubit stabilizer codespace and one or more of that stabilizer code's error spaces.
- Modular-qudit cluster-state code[674] a.k.a. Modular-qudit graph-state code.A code based on a modular-qudit cluster state.
- Modular-qudit code a.k.a. \(\mathbb{Z}_q\)-qudit code, Modular-qudit subspace code.Encodes \(K\)-dimensional Hilbert space into a \(q^n\)-dimensional (\(n\)-qudit) Hilbert space, with canonical qudit states \(|k\rangle\) labeled by elements \(k\) of the group \(\mathbb{Z}_q\) of integers modulo \(q\). Usually denoted as \(((n,K))_{\mathbb{Z}_q}\) or \(((n,K,d))_{\mathbb{Z}_q}\), whenever the code's distance \(d\) is defined, and with \(q=p\) when the dimension is prime.
- Modular-qudit color code[675] Extension of the color code to lattices of modular qudits. Codes are defined analogous to qubit color codes on suitable lattices of any spatial dimension, but a directionality is required in order to make the modular-qudit stabilizer commute. This can be done by puncturing a hyperspherical lattice [22] or constructing a star-bipartition; see [675; Sec. III]. Logical dimension is determined by the genus of the underlying surface (for closed surfaces), types of boundaries (for open surfaces), and/or any twist defects present.
- Modular-qudit stabilizer code[676] An \(((n,K,d))_q\) modular-qudit code whose logical subspace is the joint eigenspace of commuting qudit Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace, and the stabilizer group does not contain \(e^{i \phi} I\) for any \(\phi \neq 0\). The distance \(d\) is the minimum weight of a qudit Pauli string that implements a nontrivial logical operation in the code.
- Modular-qudit subsystem color code[675] An extension of subsystem color codes to modular qudits. Codes are defined analogous to qubit subsystem color codes, but a directionality is required in order to make the modular-qudit stabilizer commute [675; Sec. VII].
- Modular-qudit surface code[35,417,677] a.k.a. \(\mathbb{Z}_q\) surface code.Extension of the surface code to prime-dimensional [35,417] and more general modular qudits [677]. Stabilizer generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface. Since qudits have more than one \(X\) and \(Z\)-type operator, various sets of stabilizer generators can be defined. Ground-state degeneracy and the associated phase depends on the qudit dimension and the stabilizer generators.
- Modulation scheme
- Molecular code[263] Encodes finite-dimensional Hilbert space into the Hilbert space of \(L^2\)-normalizable functions on the group \(SO_3\). Construction is based on nested subgroups \(H\subset K \subset SO_3\), where \(H,K\) are finite. The \(|K|/|H|\)-dimensional logical subspace is spanned by basis states that are equal superpositions of elements of cosets of \(H\) in \(K\).
- Monitored random-circuit code[680–682] Error-correcting code arising from a monitored random circuit. Such a circuit is described by a series of intermittant random local projective Pauli measurements with random unitary time-evolution operators. An important sub-family consists of Clifford monitored random circuits, where unitaries are sampled from the Clifford group [683]. When the rate of projective measurements is independently controlled by a probability parameter \(p\), there can exist two stable phases, one described by volume-law entanglement entropy and the other by area-law entanglement entropy. The phases and their transition can be understood from the perspective of quantum error correction, information scrambling, and channel capacities [684,685].
- Monolithic quantum code A code constructed in a single quantum system, i.e., a physical space that is not treated as a tensor product of \(n\) identical subsystems. Examples include codes in a single qudit, spin, oscillator, or molecule.
- Movassagh-Ouyang Hamiltonian code[686] This is a family of codes derived via an algorithm that takes as input any binary classical code and outputs a quantum code (note that this framework can be extended to \(q\)-ary codes). The algorithm is probabalistic but succeeds almost surely if the classical code is random. An explicit code construction does exist for linear distance codes encoding one logical qubit. For finite rate codes, there is no rigorous proof that the construction algorithm succeeds, and approximate constructions are described instead.
- Multi-channel group-orbit code[687] Extension of binary group-orbit codes to multi-antenna transmission.
- Multi-edge LDPC code[688] Irregular LDPC code whose construction generalizes those of the original examples of irregular LDPC as well as RA codes.
- Multi-fusion string-net code[689] Family of codes resulting from the string-net construction but whose input is a unitary multi-fusion category (as opposed to a unitary fusion category).
- Multiplicity code[690–692] A generalization of an \(m\)-variate polynomial evaluation code based on evaluating polynomials and \(s\) of their derivatives at all points in \(GF(q)^m\). Originally proposed for coding using the Rosenbloom-Tsfasman metric [690]. Univariate (\(m=1\)) [690,691] and multivariante (\(m>1\)) [692] codes have been proposed.
- NTRU-GKP code[693] Multi-mode GKP code whose underlying lattice is utilized in variations of the NTRU cryptosystem [694]. Randomized constructions yield constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability.
- Nadler code[695] A nonlinear \((12,35,5)\) binary code that is the largest double-error-correcting code.
- Narrow-sense RS code[499,696,697] An \([q-1,k,n-k+1]_q\) RS code whose points \(\alpha_i\) are all \((i-1)\)st powers of a primitive element \(\alpha\) of \(GF(q)\).
- Nearly perfect code[698–700] A type of binary code whose parameters satisfy the Johnson bound with equality.
- Neural network code[701–703] An approximate code obtained from a numerical optimization involving a reinforcement learning agent.
- Newman-Moore code[704] Member of a family of \([L^2,O(L),O(L^{\frac{\log 3}{\log 2}})]\) binary linear codes on \(L\times L\) square lattices that form the ground-state subspace of a class of exactly solvable spin-glass models with three-body interactions. The codewords resemble the Sierpinski triangle on a square lattice, which can be generated by a cellular automaton [705].
- Niederreiter-Rosenbloom-Tsfasman (NRT) code[690,706–708] A poset code based on the total ordering of \([n]\), i.e., \(1\leq 2\leq \cdots \leq n\).
- Niemeier lattice code[709] One of the 24 positive-definite even unimodular lattices of rank 24.
- Niset-Andersen-Cerf code[710] Coherent-state c-q code encoding two-mode coherent states \(\{|\alpha\rangle, |\beta\rangle\}\) into four modes such that the complex values \((\alpha,\beta)\) are recoverable after a single-mode erasure. There are two variations of the storage procedure: a deterministic protocol that offers recovery against a single mode erasure, and a probabalistic that can protect against multiple errors with post selection. This code is effectively protecting classical information stored in \((\alpha,\beta)\) using quantum operations.
- Nonlinear AG code[711–715] Nonlinear \(q\)-ary code constructed by evaluating functions on an algebraic curve.
- Nordstrom-Robinson (NR) code[716,717] A nonlinear \((16,256,6)\) binary code that is the smallest Kerdock and the smallest Preparata code. The size of this code is larger than the largest possible linear code with the same length and distance.
- Norm-trace code[718] Evaluation AG code of rational functions evaluated on points lying on a Miura-Kamiya curve in either affine or projective space. The family is named as such because the equations defining the curves can be expressed in terms of the field norm and field trace.
- Number-phase code[130] a.k.a. Single-mode translationally invariant Fock-state code.Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states [719], \begin{align} |\phi\rangle \equiv \frac{1}{\sqrt{2\pi}}\sum_{n=0}^{\infty} \mathrm{e}^{\mathrm{i} n \phi} \ket{n}. \tag*{(11)}\end{align} Since phase states and thus the ideal codewords are not normalizable, approximate versions need to be constructed. The codes' key feature is that, in the ideal case, phase measurement has zero uncertainty, making it a good canditate for a syndrome measurement.
- Numerically optimized bosonic code[123,124] Bosonic Fock-state code obtained from a numerical minimization procedure, e.g., from enforcing error-correction criteria against some number of losses while minimizing average occupation number. Useful single-mode codes can be determined using basic numerical optimization [123,124], semidefinite-program recovery/encoding optimization [720,721], or reinforcement learning [722,723].
- Octacode[110,724,725] The unique self-dual linear code of length 8 and Lee distance 6 over \(\mathbb{Z}_4\) with generator matrix \begin{align} \begin{pmatrix} 3 & 3 & 2 & 3 & 1 & 0 & 0 & 0\\ 3 & 0 & 3 & 2 & 3 & 1 & 0 & 0\\ 3 & 0 & 0 & 3 & 2 & 3 & 1 & 0\\ 3 & 0 & 0 & 0 & 3 & 2 & 3 & 1 \end{pmatrix}\,. \tag*{(12)}\end{align}
- On-off keyed (OOK) c-q code[726] Coherent-state c-q binary code whose encoding is either in the vacuum \(|0\rangle\) or in a nonzero coherent state \(|\alpha\rangle\).
- One-hot code[727] a.k.a. One-vs-all (OVA) code, One-against-all (1AA) code, One-vs-rest (OvR) code, \(1\)-in-\(n\) code.A length-\(n\) binary code whose codewords are those with Hamming weight one. The reverse of this code, where all codewords have Hamming weight \(n-1\) is called a one-cold code.
- One-hot quantum code[728] a.k.a. Single-excitation subspace code, Direct mapping, Multi-rail code.Encoding of a \(q\)-dimensional qudit into the single-excitation subspace of \(q\) modes. The \(j\)th logical state is the multi-mode Fock state with one photon in mode \(j\) and zero photons in the other modes. This code is useful for encoding and performing operations on qudits in multiple qubits [729–733].
- One-versus-one (OVO) code[734,735] a.k.a. One-against-one (1A1) code.A length-\(n\) ternary code over \(\{\om 1,0\}\) whose whose generator matrix has columns with one \(+1\), one \(-1\), and with the rest of the entries zero.
- Operator-algebra (OA) qubit code An OAQECC family that encompasses ordinary (i.e., subspace) qubit codes, subsystem qubit codes, and hybrid qubit codes using a unified operator-algebraic framework.
- Operator-algebra (OA) qubit stabilizer code[736] An OAQECC in which the commutant \(\mathcal{A}'\) of the logical algebra \(\mathcal{A}\) arises as the group algebra of a subgroup \(\mathsf{G}\) of the \(n\)-qubit Pauli group \(\mathsf{P}_n\).
- Operator-algebra QECC (OAQECC)[533,737–741] A code family that encompasses ordinary (i.e., subspace) codes, subsystem codes, classical-quantum codes, and hybrid codes using a unified operator-algebraic framework.
- Optimal LRC[742,743] An LRC whose parameters saturate a generalized Singleton bound.
- Orthogonal Spacetime Block Code (OSTBC)[38] The codewords are \(T\times n\) matrices as defined for spacetime codes, with the additional condition that columns of the coding matrix are orthogonal. The parameter \(n\) is the number of channels, and \(T\) is the number of time slots.
- Orthogonal array (OA)[744–746] An orthogonal array, or OA\(_{\lambda}(t,n,q)\), of strength \(t\) with \(q\) levels and \(n\) constraints is a set of \(q\)-ary strings such that any subset of \(t\) coordinates contains every length-\(t\) string an equal number of times \(\lambda\), which is the index of the array.
- Oscillator-into-oscillator GKP code[747] a.k.a. GKP-stabilizer code.Multimode GKP code with an infinite-dimensional logical space. Can be obtained by considering an \(n\)-mode GKP code with a finite-dimensional logical space, removing stabilizers such that the logical space becomes infinite dimensional, and applying a Gaussian circuit.
- Oscillator-into-oscillator code[748,749] a.k.a. Analog quantum code.Encodes \(k\) bosonic modes into \(n\) bosonic modes.
- Ouyang-Chao constant-excitation PI code[750] A constant-excitation PI Fock-state code whose construction is based on integer partitions.
- Ovoid code[555,751] Member of a \([q^2+1,4,q^2-q]_q\) projective code family that is universally optimal and that is constructed using ovoids in projective space. See [752; pg. 107][70; pg. 192] for further details.
- PI qubit code Block quantum code defined on two-dimensional subsystems such that any permutation of the subsystems leaves any codeword invariant.
- PPM c-q code[753] A \(q\)-PPM c-q code is a coherent-state c-q code whose \(j\)th codeword corresponds to a tensor-product state of zero-amplitude coherent states at all modes except mode \(j\). For example, a 3-PPM encoding corresponds to the three-mode states \(|\alpha\rangle|0\rangle|0\rangle\), \(|0\rangle|\alpha\rangle|0\rangle\), and \(|0\rangle|0\rangle|\alpha\rangle\) for some complex \(\alpha\). The dual of a PPM code is obtained by the exchange \(0\leftrightarrow\alpha\).
- PSK c-q code[754] Coherent-state c-q \(q\)-ary code whose \(j\)th codeword corresponds to a coherent state whose phase is the \(j\)th multiple of \(2\pi/q\). These states are also called geometrically uniform states (GUS) [755].
- Pair-cat code[756] Two- or higher-mode extension of cat codes whose codewords are right eigenstates of powers of products of the modes' lowering operators. Many gadgets for cat codes have two-mode pair-cat analogues, with the advantage being that such gates can be done in parallel with a dissipative error-correction process.
- Parallel concatenated code A code that is constructed by combining two or more codes in a Tanner code, in a tensor-product code, or in a modified Tanner construction [757].
- Parallel-recovery code[758] A \(t\)-erasure LRC whose coordinate erasures can be recovered in parallel.
- Parvaresh-Vardy (PV) code[759] a.k.a. Correlated RS code.An IRS code with additional algebraic relations (a.k.a. correlations) between the codeword polynomials \(\{f^{(j)}\}_{j=1}^{t}\). These relations yielded a list decoder that achieves list-decoding capacity.
- Pastawski-Yoshida-Harlow-Preskill (HaPPY) code[510] a.k.a. Perfect holographic code.Holographic code constructed out of a network of hexagonal perfect tensors that tesselates hyperbolic space. The code serves as a minimal model for several aspects of the AdS/CFT holographic duality [760] and potentially a dF/CFT duality [761]. It has been generalized to higher dimensions [762] and to include gauge-like degrees of freedom on the links of the tensor network [763,764]. All boundary global symmetries must be dual to bulk gauge symmetries, and vice versa [765].
- Penrose tiling code[766] Encodes quantum information into superpositions of rotated and translated versions of different Penrose tilings of \(\mathbb{R}^n\).
- Pentacode[767] Nonlinear \((5,40,4)_{\mathbb{Z}_4}\) code over \(\mathbb{Z}_4\) whose codewords are all cyclic permutations and negations of the strings \(01112\), \(03110\), \(21310\), and \(21132\).
- Perfect binary code An \((n,K,2t+1)\) binary code is perfect if parameters \(n\), \(K\), and \(t\) are such that the binary Hamming (a.k.a. sphere-packing) bound \begin{align} \sum_{j=0}^{t} {n \choose j} \leq 2^{n}/K \tag*{(13)}\end{align} becomes an equality. For example, for a code with one logical bit (\(K=2\)) and \(t=1\), the bound becomes \(n+1 \leq 2^{n-1}\). Perfect codes are those for which balls of Hamming radius \(t\) exactly fill the space of all \(n\) binary strings.
- Perfect code A type of \(q\)-ary code whose parameters satisfy the Hamming bound with equality.
- Perfect quantum code A type of block quantum code whose parameters satisfy the quantum Hamming bound with equality.
- Perfect-tensor code a.k.a. AME code.Block quantum code encoding one subsystem into \(n\) subsystems whose encoding isometry is a perfect tensor. This code stems from an AME\((n,q)\) AME state, or equivalently, a \(((n+1,1,\lfloor (n+1)/2 \rfloor + 1))_{\mathbb{Z}_q}\) code.
- Permutation spherical code[768,769] Slepian group-orbit code whose codewords are constructed from an arbitrary unit vector in two possible variants. Variant 1 consists of codewords that are permutations of the vector's coordinates, while Variant 2 consists of such permutations and all possible sign changes of the vector's components.
- Permutation-invariant (PI) code[770] Block quantum code such that any permutation of the subsystems leaves any codeword invariant. In other words, the automorphism group of the code contains the symmetric group \(S_n\).
- Petersen cycle code[238] A \([15,6,5]\) cycle code whose parity-check matrix is the incidence matrix of the Petersen graph. The Petersen graph can be thought of as a dodecahedron with antipodes identified [771; Appx. A.2.1].
- Petersen spherical code[772] A \((4,10,1/6)\) spherical code whose codewords correspond to vertices of the Peterson graph. Its Gram matrix is constructed by putting \(-2/3\) whenever two vertices are adjacent in the graph, and \(1/6\) otherwise. The code is optimal for its parameters [772].
- Phase-shift keying (PSK) code A \(q\)-ary phase-shift keying (\(q\)-PSK) encodes one \(q\)-ary digit of information into a constellation of \(q\) points distributed equidistantly on a circle in \(\mathbb{C}\) or, equivalently, \(\mathbb{R}^2\).
- Planar-perfect-tensor code[489,773] a.k.a. Block-perfect-tensor code, Perfect-tangle code.Block quantum code whose encoding isometry is a block perfect tensor, i.e., a tensor which remains an isometry under partitions into two contiguous components in a fixed plane. This code stems from a planar maximally entangled state [774].
- Plane-curve code[775] Evaluation AG code of bivariate polynomials of some finite maximum degree, evaluated at points lying on an affine or projective plane curve.
- Pless symmetry code[776,777] a.k.a. Pless double circulant code.A member of a family of self-dual ternary \([2q+2,q+1]_3\) codes for any power of an odd prime satisfying \(q \equiv 2\) modulo 3.
- Polar c-q code[778,779] Polar code adapted to transmit classical information over channels with classical inputs and quantum outputs.
- Polar code[780] In its basic version, a binary linear polar code encodes \(K\) message bits into \(N=2^n\) bits. The linear transformation that defines the code is given by the matrix \(G^{(n)}=B_N G^{\otimes n}\), where \(B_N\) is a certain \(N\times N\) permutation matrix, and \(G^{\otimes n}\) is the \(n\)th Kronecker power of the \(2\times 2\) kernel matrix \(G=\left[\begin{smallmatrix}1 & 0\\ 1 & 1 \end{smallmatrix}\right]\). To encode \(K\) message bits, one forms an \(N\)-vector \(u\) in which \(K\) coordinates represent the message bits. The remaining \(N-K\) coordinates are set to some fixed values and are said to be frozen. The codeword \(x \in \{0,1\}^N\) is obtained as \(x=u G^{\otimes n}\).
- Polygon code Spherical \((1,q,4\sin^2 \frac{\pi}{q})\) code for any \(q\geq1\) whose codewords are the vertices of a \(q\)-gon. Special cases include the line segment (\(q=2\)), triangle (\(q=3\)), square (\(q=4\)), pentagon (\(q=5\)), and hexagon (\(q=6\)).
- Polynomial evaluation code Evaluation code of polynomials (or, more generally, rational functions) at points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) on an algebraic variety \(\cal X\) of dimension greater than one (i.e., not an algebraic curve).
- Polyphase code[781–791] A spherical code obtained from a binary code, \(q\)-ary code, or \(q\)-ary code over \(\mathbb{Z}_q\) via a component-wise mapping of each \(q\)-ary digit to a \(q\)th root of unity.
- Polytope code Spherical code whose codewords are the vertices of a polytope, i.e., a geometrical figure bounded by lines, planes, and hyperplanes [3]. Polytopes in two (three) real or complex dimensions are called polygons (polyhedra).
- Poset code[792] Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\), with its distance evaluated in the poset metric.
- Preparata code[793] A nonlinear binary \((2^{m+1}-1, 2^{m+1}-2m-2, 5)\) code where \(m\) is odd. The size of this code is twice the size of the largest possible linear code with the same length and distance.
- Prime-qudit RM code[413,794] Modular-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes or their duals via the modular-qudit CSS construction. An odd-prime-qudit CSS code family constructed from first-order punctured GRM codes transversally implements a diagonal gate at any level of the qudit Clifford hierarchy [794].
- Prime-qudit RS code[795] a.k.a. Prime-qudit polynomial code (QPyC).Prime-qudit CSS code constructed using two RS codes.
- Prime-qudit triorthogonal code[796] An \(m \times n\) matrix over \(GF(p)=\mathbb{Z}_p\) is triorthogonal if its rows \(r_1, \ldots, r_m\) satisfy \(|r_i \cdot r_j| = 0\) and \(|r_i \cdot r_j \cdot r_k| = 0\) modulo \(p\), where addition and multiplication are done on \(GF(p)\). The triorthogonal prime-qudit CSS code associated with the matrix is constructed by mapping non-zero entries in self-orhogonal rows to \(X\) operators, and \(Z\) operators for each row in the orthogonal complement [796,797].
- Primitive narrow-sense BCH code[127] BCH codes for \(b=1\) and for \(n=q^r-1\) for some \(r\geq 2\).
- Private information retrieval (PIR) code[798,799] A code used to obtain information from several servers privately, i.e., without the servers knowing what information was obtained.
- Product-matrix (PM) code[800] Code constructed using two explicit constructions, with each construction corresponding to one of the two extreme points of the storage-bandwidth trade-off curve [801].
- Projective RM (PRM) code[802,803] Reed-Muller code for nonzero points \(\{\alpha_1,\cdots,\alpha_n\}\) with \(n=m+1\) whose leftmost nonzero coordinate is one, corresponding to an evaluation code of polynomials over projective coordinates.
- Projective geometry code Linear \(q\)-ary \([n,k,d]\) code such that columns of its generator matrix \(G\) does not contain any repeated columns or the zero column. That way, each column corresponds to a distinct point in the projective space \(PG(k-1,q)\) arising from a \(k\)-dimensional vector space over \(GF(q)\). If the columns are linearly independent, then the codewords are collectively called an information set. Columns of a code's parity-check matrix can similarly correspond to points in projective space. This formulation yields connections to projective geometry, which can be applied to determine code properties.
- Projective two-weight code A projective code whose codewords all have one of two possible nonzero Hamming weights.
- Projective-plane surface code[804] A family of Kitaev surface codes on the non-orientable 2-dimensional compact manifold \(\mathbb{R}P^2\) (in contrast to a genus-\(g\) surface). Whereas genus-\(g\) surface codes require \(2g\) logical qubits, qubit codes on \(\mathbb{R}P^2\) are made from a single logical qubit.
- Protograph LDPC code[805–807] Binary version of a \(q\)-ary protograph LDPC code. Its parity check matrix can be put into the form of a block matrix consisting of either a sum of permutation sub-matrices or the zero sub-matrix.
- Pulse-amplitude modulation (PAM) code Encodes a \(q\)-ary digit into a constellation of equally spaced points on the real line. For example, a \(q\)-PAM scheme for \(q=8\) could encode the constellation \(\{ \pm \alpha,\pm 3\alpha,\pm 5\alpha, \pm 7\alpha \}\) with real scaling factor \(\alpha\). The points in the constellation are typically associated with one quadrature of an electromagnetic signal.
- Pulse-position modulation (PPM) code An analog code encoding into \(q\) different signals such that each codeword corresponds to a signal.
- Purity-testing stabilizer code[808] A qubit stabilizer code that is constructed from a normal rational curve and that is relevant to testing the purity of an entangled Bell state stabilized by two parties [808].
- Pyramid code[809] An LRC whose generator matrix is that of an RS code in standard form, but some of whose columns are split into multiple columns; see [185; Sec. 31.3.1.1] for an example.
- Quadrature PSK (QPSK) code[810] a.k.a. Quadriphase PSK code, 4-PSK code, 4-QAM code.A quaternary encoding into a constellation of four points distributed equidistantly on a circle. For the case of \(\pi/4\)-QPSK, the constellation is \(\{e^{\pm i\frac{\pi}{4}},e^{\pm i\frac{3\pi}{4}}\}\).
- Quadrature-amplitude modulation (QAM) code Encodes into points into a subset of points lying on in \(\mathbb{R}^{2}\), here treated as \(\mathbb{C}\). Each pair of points is associated with a complex amplitude of an electromagnetic signal, and information is encoded into both the norm and phase of that signal [679; Ch. 16].
- Quadric code[811,812] Evaluation code of polynomials evaluated on points lying on a quadric hypersurface.
- Quantum AG code[813] A Galois-qudit CSS code constructed using two linear AG codes.
- Quantum Golay code[82] a.k.a. Qubit Golay code.A \([[23, 1, 7]]\) self-dual CSS code with eleven stabilizer generators of each type, and with each generator being weight eight.
- Quantum Goppa code[814–816] A Galois-qudit CSS code constructed using two Goppa codes.
- Quantum LDPC (QLDPC) code a.k.a. Sparse quantum code.Member of a family of \([[n,k,d]]\) modular-qudit or Galois-qudit stabilizer codes for which the number of sites participating in each stabilizer generator and the number of stabilizer generators that each site participates in are both bounded by a constant \(w\) as \(n\to\infty\); can be denoted by \([[n,k,d,w]]\). Sometimes, the two parameters are explicitly stated: each site of an an \((l,w)\)-regular QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\). QLDPC codes can correct many stochastic errors far beyond the distance, which may not scale as favorably. Together with more accurate, faster, and easier-to-parallelize measurements than those of general stabilizer codes, this property makes QLDPC codes interesting in practice.
- Quantum Reed-Muller code[817,818] A CSS code formed from a classical Reed-Muller (RM) code or its punctured/shortened versions. Such codes often admit transversal logical gates in the Clifford hierarchy.
- Quantum Tamo-Barg (QTB) code[819] A member of a family of Galois-qudit CSS codes whose underlying classical codes consist of Tamo-Barg codes together with specific low-weight codewords. Folded versions of QTB codes, or FQTB codes, defined on qudits whose dimension depends on \(n\) yield explicit examples of QLRCs of arbitrary locality \(r\) [819; Thm. 2].
- Quantum Tanner code[820] Member of a family of QLDPC codes based on two compatible classical Tanner codes defined on a two-dimensional Cayley complex, a complex constructed from Cayley graphs of groups. For certain choices of codes and complex, the resulting codes have asymptotically good parameters. This construction has been generalized to Schreier graphs [437].
- Quantum check-product code[821] CSS code constructed from an extension of check product (between two classical codes) to a product between a classical and a quantum code.
- Quantum code Code designed for transmission of quantum and/or classical information through a quantum channel for the purposes of robust storage, communication, or sensing. Transmission can be performed with side information or entanglement.
- Quantum convolutional code[822,823] One-dimensional translationally invariant qubit stabilizer code whose whose stabilizer group can be partitioned into constant-size subsets of constant support and of constant overlap between neighboring sets. Initially formulated as a quantum analogue of convolutional codes, which were designed to protect a continuous and never-ending stream of information. Precise formulations sometimes begin with a finite-dimensional lattice, with the intent to take the thermodynamic limit; logical dimension can be infinite as well.
- Quantum data-syndrome (QDS) code[824–828] Stabilizer code designed to correct both data qubit errors and syndrome measurement errors simultaneously due to extra redundancy in its stabilizer generators.
- Quantum divisible code[829–831] A level-\(\nu\) quantum divisible code is a CSS code whose \(X\)-type stabilizers form a \(\nu\)-even linear binary code in the symplectic representation and which admits a transversal gate at the \(\nu\)th level of the Clifford hierarchy. A CSS code is doubly even (triply even) if all \(X\)-type stabilizers have weight divisible by four (eight), i.e., if they form a doubly even (triply even) linear binary code.
- Quantum duadic code[832–835] True Galois-qudit stabilizer code constructed from \(q\)-ary duadic codes via the Hermitian construction or the Galois-qudit CSS construction.
- Quantum error-correcting code (QECC) Encodes quantum information in a (logical) subspace of a (physical) Hilbert space such that it is possible to recover said information from errors that act as linear maps on the physical space.
- Quantum error-transmuting code (QETC)[836] Encodes quantum information in a (logical, \(k\)-qubit) subspace \(\mathsf{C}\) of a (physical, \(n\)-qubit) Hilbert space \(\mathsf{H}\) such that recovery is possible from a set of physical errors occurring up to a pre-specified (smaller, but non-empty) admissible set of logical errors. This is relevant to, e.g., simulation of noisy systems. Most QETCs are stabilizer codes: \(\mathsf{C}\) is the subspace stabilised by an abelian subgroup \(\mathsf{S} \subset \mathcal{G}_n\) of the Pauli group on \(n\) qubits.
- Quantum expander code[837] CSS codes constructed from a hypergraph product of bipartite expander graphs [328] with bounded left and right vertex degrees. For every bipartite graph there is an associated matrix (the parity check matrix) with columns indexed by the left vertices, rows indexed by the right vertices, and 1 entries whenever a left and right vertex are connected. This matrix can serve as the parity check matrix of a classical code. Two bipartite expander graphs can be used to construct a quantum CSS code (the quantum expander code) by using the parity check matrix of one as \(X\) checks, and the parity check matrix of the other as \(Z\) checks.
- Quantum lattice code Bosonic stabilizer code on \(n\) bosonic modes whose stabilizer group is an infinite countable group of oscillator displacement operators which implement lattice translations in phase space.
- Quantum locally recoverable code (QLRC)[819] A QLRC of locality \(r\) is a block quantum code whose code states can be recovered after a single erasure by performing a recovery map on at most \(r\) subsystems.
- Quantum locally testable code (QLTC)[838] A local commuting-projector Hamiltonian-based block quantum code which has a nonzero average-energy penalty for creating large errors. Informally, QLTC error states that are far away from the codespace have to be excited states by a number of the code's local projectors that scales linearly with \(n\).
- Quantum low-weight check (QLWC) code[839] Member of a family of \([[n,k,d]]\) modular-qudit or Galois-qudit stabilizer codes for which the number of sites participating in each stabilizer generator is bounded by a constant as \(n\to\infty\).
- Quantum maximum-distance-separable (MDS) code[840–842] A type of block quantum code whose parameters satisfy the quantum Singleton bound with equality.
- Quantum multi-dimensional parity-check (QMDPC) code[843] High-rate low-distance CSS code whose qubits lie on a \(D\)-dimensional rectangle, with \(X\)-type stabilizer generators defined on each \(D-1\)-dimensional rectangle. The \(Z\)-type stabilizer generators are defined via permutations in order to commute with the \(X\)-type generators.
- Quantum parity code (QPC)[91,844,845] a.k.a. Subspace Shor code.A \([[m_1 m_2,1,\min(m_1,m_2)]]\) CSS code family obtained from concatenating an \(m_1\)-qubit phase-flip repetition code with an \(m_2\)-qubit bit-flip repetition code.
- Quantum pin code[544] Member of a family of CSS codes that encompasses both quantum Reed-Muller and color codes and that is defined using intersections of pinned sets.
- Quantum polar code[846] Entanglement-assisted CSS code utilized in a quantum polar coding scheme producing entangled pairs of qubits between sender and receiver. In such a scheme, the amplitude and phase information of a quantum state is handled in complementary fashion [847] using an encoding based on classical polar codes. Variants of the initial scheme have been developed for degradable channels [848] and extended to arbitrary channels [849].
- Quantum quadratic-residue (QR) code[414,416,842] Galois-qudit \([[n,1]]_q\) pure self-dual CSS code constructed from a dual-containing QR code via the Galois-qudit CSS construction. For \(q\) not divisible by \(n\), its distance satisfies \(d^2-d+1 \geq n\) when \(n \equiv 3\) modulo 4 [416; Thm. 40] and \(d \geq \sqrt{n}\) when \(n\equiv 1\) modulo 4 [416; Thm. 41].
- Quantum rainbow code[850] A CSS code whose qubits are associated with vertices of a simplex graph with \(m+1\) colors.
- Quantum repetition code[851] Encodes \(1\) qubit into \(n\) qubits according to \(|0\rangle\to|\phi_0\rangle^{\otimes n}\) and \(|1\rangle\to|\phi_1\rangle^{\otimes n}\). The code is called a bit-flip code when \(|\phi_i\rangle = |i\rangle\), and a phase-flip code when \(|\phi_0\rangle = |+\rangle\) and \(|\phi_1\rangle = |-\rangle\).
- Quantum spatially coupled (SC-QLDPC) code[852,853] QLDPC code whose stabilizer generator matrix resembles the parity-check matrix of SC-LDPC codes. There exist CSS [852] and stabilizer constructions [853]. In either case, the stabilizer generator matrix is constructed by "spatially" coupling sub-matrix blocks in chain-like fashion (or, more generally, in grid-like fashion) to yield a band matrix. The sub-matrix blocks have to satisfy certain conditions amongst themselves so that the resulting band matrix is a stabilizer generator matrix. Matrices corresponding to translationally invariant chains are called time-variant, and otherwise are called time-invariant.
- Quantum spherical code (QSC)[178] Code whose codewords are superpositions of points on an \(n\)-dimensional real or complex sphere. Such codes can in principle be defined on any configuration space housing a sphere, but the focus of this entry is on QSCs constructed out of coherent-state constellations.
- Quantum tensor-product code[854,855] CSS code constructed from a tensor code. In some cases, only one of the classical codes forming the tensor code needs to be self-orthogonal.
- Quantum turbo code[856,857] A quantum version of the turbo code, obtained from an interleaved serial quantum concatenation [857; Def. 30] of quantum convolutional codes.
- Quantum twisted code[490] Hermitian code arising constructed from twisted BCH codes.
- Quantum-double code[35] Group-GKP stabilizer code whose codewords realize 2D modular gapped topological order defined by a finite group \(G\). The code's generators are few-body operators associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit of dimension \( |G| \) located at each edge of the tesselation).
- Quantum-inspired classical block code A block code of length \(n\) whose construction was inspired by a quantum code.
- Quasi group-algebra code a.k.a. Quasi-\(G\) code.A \(q\)-ary linear code based on a finite group \( G \) of order \(n/\ell\) for some index \(\ell\). The code is a right submodule of the direct sum of \(\ell\) copies of the group algebra \(\mathbb{F}_q G\). A quasi group-algebra code for an Abelian group is called an Abelian quasi group-algebra code.
- Quasi-cyclic LDPC (QC-LDPC) code[48,858–863][386; Appx. C] LDPC code that can be put into quasi-cyclic form. Its parity check matrix can be put into the form of a block matrix consisting of either circulant permutation sub-matrices or the zero sub-matrix. Such codes are often constructed by lifting certain protographs into such block matrices [864]. Their simple structure makes them useful for several wireless communication standards.
- Quasi-cyclic QLDPC code[865,866]
- Quasi-cyclic code[867] A block code of length \(n\) is quasi-cyclic if, for each codeword \(c_1 \cdots c_{\ell} c_{\ell+1} \cdots c_n\), the string \(c_{n-\ell+1} \cdots c_n c_1 \cdots c_{n-\ell}\), where each entry is cyclically shifted by \(\ell\) increments, is also a codeword.
- Quasi-cyclic quantum code[865] A block code on \(n\) subsystems such that cyclic shifts of the subsystems by \(\ell\geq 1\) leave the codespace invariant.
- Quasi-hyperbolic color code[868] An extension of the color code construction to quasi-hyperbolic manifolds, e.g., a product of a 2D hyperbolic surface and a circle.
- Quasi-perfect code Perfect codes \((n,K,d)_q\) are those for which balls of Hamming radius \(t=\left\lfloor (d-1)/2\right\rfloor\) exactly fill the space of all \(n\) \(q\)-ary strings. Quasi-perfect codes are those for which balls of Hamming radius \(t\) are disjoint, while balls of radius \(t+1\) cover the space with possible overlaps. In other words, any \(q\)-ary string is at most \(t+1\) bit flips away from a codeword of a quasi-perfect code.
- Quasi-twisted code A block code of length \(n\) is \(\alpha\)-quasi-twisted if, for each codeword \(c_1 \cdots c_{\ell} c_{\ell+1} \cdots c_n\), the string \(\alpha c_{n-\ell+1}, \alpha c_{n-\ell+2}, \cdots, \alpha c_n, c_1, c_2, \cdots, c_{n-\ell}\) is also a codeword.
- Quaternary RM (QRM) code[869] A quaternary linear code over \(\mathbb{Z}_4\) that is a quaternary version of the RM code in that its binary image under the Gray map is an RM code. This code subsumes the quaternary images of the Kerdock and Preparata codes under the Gray map. The code is usually noted as QRM\((r,m)\), with its image under the Gray map yielding the RM code RM\((r,m)\) [869; Thm. 19].
- Quaternary linear code over \(\mathbb{Z}_4\) A linear code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_4\) of integers modulo 4.
- Qubit BCH code[82,282,870–872] Qubit stabilizer code constructed from a self-orthogonal binary BCH code via the CSS construction, from a Hermitian self-orthogonal quaternary BCH code via the Hermitian construction, or by taking a Euclidean self-orthogonal BCH code over \(GF(2^m)\), converting it to a binary code, and applying the CSS construction.
- Qubit CSS code[395–397] a.k.a. Qubit Euclidean code.An \([[n,k,d]]\) stabilizer code admitting a set of stabilizer generators that are either \(Z\)-type or \(X\)-type Pauli strings. Codes can be defined from two classical codes and/or chain complexes over \(\mathbb{Z}_2\) per the qubit CSS-to-homology correspondence below. Strong CSS codes are codes for which there exists a set of \(X\) and \(Z\) stabilizer generators of equal weight.
- Qubit c-q code Qubit code designed for transmission of classical information in the form of bits through non-classical channels.
- Qubit code a.k.a. Qubit subspace code.Encodes \(K\)-dimensional Hilbert space into a \(2^n\)-dimensional (i.e., \(n\)-qubit) Hilbert space. Usually denoted as \(((n,K))\) or \(((n,K,d))\), where \(d\) is the code's distance.
- Qubit stabilizer code[676,873] a.k.a. Pauli stabilizer code, Additive quantum code, Additive CWS code, Clifford code.An \(((n,2^k,d))\) qubit stabilizer code is denoted as \([[n,k]]\) or \([[n,k,d]]\), where \(d\) is the code's distance. Logical subspace is the joint eigenspace of commuting Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace of a set of \(2^{n-k}\) commuting Pauli operators which do not contain \(-I\). The distance is the minimum weight of a Pauli string that implements a nontrivial logical operation in the code.
- Qudit GNU PI code[874] Extension of the GNU PI codes to those encoding logical qudits into physical qubits. Codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of polynomial coefficients, with the case of binomial coefficients reducing to the GNU PI codes.
- Qudit cubic code[875–877] Generalization of the Haah cubic code to modular qudits.
- Qudit-into-oscillator code Encodes \(K\)-dimensional Hilbert space into \(n\) bosonic modes.
- RS NRT code[690] An NRT analogue of an RS code.
- Random code[337] Code whose construction is non-deterministic in some way, i.e., codes that utilize an elements of randomness somewhere in their construction. Members of this class range from fully non-deterministic codes, to codes whose multi-step construction is deterministic with the exception of a single step.
- Random quantum code Quantum code whose construction is non-deterministic in some way, i.e., codes that utilize an elements of randomness somewhere in their construction. Members of this class range from fully non-deterministic codes (e.g., random-circuit codes), to codes whose multi-step construction is deterministic with the exception of a single step (e.g., expander lifter-product codes).
- Random stabilizer code[395,676,873] a.k.a. Random Clifford-circuit code.An \(n\)-qubit, modular-qudit, or Galois-qudit stabilizer code whose construction is non-deterministic. Since stabilizer encoders are Clifford circuits, such codes can be thought of as arising from random Clifford circuits.
- Random-circuit code[878] Code whose encoding is naturally constructed by randomly sampling from a large set of (not necessarily unitary) quantum circuits.
- Rank-metric code[383] a.k.a. Delsarte rank-metric code.Each codeword is a matrix over \(GF(q)\), with codewords forming a \(GF(q)\)-linear subspace, and with the metric being the rank of the difference of matrices. The distance \(d\) is the minimum rank of all nonzero matrices in the code. Rank-metric codes on \(n\times m\) matrices are denoted as \([n\times m,k,d]_q\).
- Rank-modulation code[879,880] A family of codes in permutations derived from \(q\)-ary linear codes, such as Lee-metric codes, RS codes [880], quadratic residue codes, and most binary codes.
- Raptor (RAPid TORnado) code[881,882] Raptor codes are concatenated erasure codes with two layers: an outer pre-code and a Luby-Transform (LT) inner code. The pre-code is a linear binary erasure code, which is applied first to the input to create some redundant data. The LT code is then applied to the intermediate symbols from the pre-code to generate final output symbols to be transmitted.
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code[883–885] a.k.a. Raussendorf-Harrington-Goyal (RHG) cluster-state code.A three-dimensional cluster-state code defined on the bcc lattice (i.e., a cubic lattice with qubits on edges and faces).
- Real-Clifford subgroup-orbit code[886,887] Slepian group-orbit code of dimension \(2^r\), approximate asympotic size \(2.38 \cdot 2^{r(r+1)/2+1}\), and distance \(1\). Code is constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group [179], onto the vector \((1,0,0,\cdots,0)\). This group is the automorphism group of BW lattice, and the resulting codes coincide with the optimal spherical codes for dimensions \(\{4,8,16\}\).
- Rectified Hessian polyhedron code Spherical \((6,72,1)\) code whose codewords are the vertices of the rectified Hessian complex polyhedron and the \(1_{22}\) real polytope. Codewords form the minimal lattice-shell code of the \(E_6\) lattice. See [888; pg. 127][110; pg. 126] for realizations of the 72 codewords.
- Reed-Muller (RM) code[889–891] Member of the RM\((r,m)\) family of linear binary codes derived from multivariate polynomials. The code parameters are \([2^m,\sum_{j=0}^{r} {m \choose j},2^{m-r}]\), where \(r\) is the order of the code satisfying \(0\leq r\leq m\). First-order RM codes are also called biorthogonal codes, while \(m\)th order RM codes are also called universe codes. Punctured RM codes RM\(^*(r,m)\) are obtained from RM codes by deleting one coordinate from each codeword.
- Reed-Solomon (RS) code[499,696,697] An \([n,k,n-k+1]_q\) linear code based on polynomials over \(GF(q)\).
- Regenerating code (RGC)[801] An \([n,k,d,\alpha,\beta,M]_q\) Regenerating Code \(\mathcal{C}\) is an erasure correcting code that encodes \(M\) symbols from \(GF(q)\) into an \(\alpha \times n\) matrix over \(GF(q)\), with each column of the matrix treated as a coordinate of a codeword.
- Regular LDPC code An LDPC code whose parity-check matrix has a fixed number of entries for each row or column.
- Regular binary Tanner code[892] a.k.a. Regular binary GLDPC code.A binary Tanner code defined on a regular bipartite graph, with the inner code being the same for all vertices.
- Renormalization group (RG) cat code[741,893,894] Code whose codespace is spanned by \(q\) field-theoretic coherent states which are flowing under the renormalization group (RG) flow of massive free fields. The code approximately protects against displacements that represent local (i.e., short-distance, ultraviolet, or UV) operators. Intuitively, this is because RG cat codewords represent non-local (i.e., long-distance) degrees of freedom, which should only be excitable by acting on a macroscopically large number of short-distance degrees of freedom.
- Repeat-accumulate (RA) code[895] An LDPC code whose parity-check matrix has weight-two columns arranged in a step-like pattern for its last columns [896].
- Repeat-accumulate-accumulate (RAA) code[897] Generalization of the RA code in which two accumulators and permutations are used.
- Repetition code \([n,1,n]\) binary linear code encoding one bit of information into an \(n\)-bit string. The length \(n\) needs to be an odd number, since the receiver will pick the majority to recover the information. The idea is to increase the code distance by repeating the logical information several times. It is a \((n,1)\)-Hamming code. Its automorphism group is \(S_n\).
- Residue AG code a.k.a. Differential code.Linear \(q\)-ary code defined using a set of points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) in \(GF(q)\) lying on an algebraic curve \(\cal X\) and a linear space \(\Omega\) of certain rational differential forms \(\omega\).
- Reversible code A code of length \(n\) over an alphabet is reversible if, for each codeword \(c_1 c_2 \cdots c_n\), the reversed string \(c_n \cdots c_2 c_1\) is also a codeword.
- Rhombic dodecahedron surface code[898] a.k.a. Landahl jaunty code.A \([[14,3,3]]\) twist-defect surface code whose qubits lie on the vertices of a rhombic dodecahedron. Its non-CSS nature is due to twist defects [899] stemming from the geometry of the polytope.
- Ring code Encodes \(K\) states (codewords) in \(n\) coordinates over a finite ring \(R\).
- Root lattice code A lattice that is symmetric under a specific crystallographic reflection group; see [110; Table 4.1] for the list of crystallographic reflection groups and their corresponding root lattices. The root-lattice family consists of lattices \(A_n\), \(\mathbb{Z}^n\), or \(D_n\) for dimension \(n\), or \(E_{i}\) for \(i\in\{6,7,8\}\). Their generator matrices can be taken to be the root matrices of the corresponding reflection groups.
- Rotated surface code[551,900–902] a.k.a. Checkerboard code, Medial surface code, Rectified surface code.Variant of the surface code defined on a square lattice that has been rotated 45 degrees such that qubits are on vertices, and both \(X\)- and \(Z\)-type check operators occupy plaquettes in an alternating checkerboard pattern.
- Roth-Lempel code[903] Member of a \(q\)-ary linear code family that includes many examples of MDS codes that are not GRS codes.
- Rotor GKP code[263,452,904] GKP code protecting against small angular position and momentum shifts of a planar rotor.
- Rotor code Encodes a logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space of \(L^2\)-normalizable functions on either the integers \(\mathbb Z\) or the circle group \(U(1)\). Ideal codewords may not be normalizable because the space is infinite-dimensional, so approximate versions have to be constructed in practice.
- Rotor stabilizer code Rotor code whose codespace is defined as the common \(+1\) eigenspace of a group of mutually commuting rotor generalized Pauli operators. The stabilizer group can be either discrete or continuous, corresponding to modular or linear constraints on angular positions and momenta. Both cases can yield finite or infinite logical dimension. Exact codewords are non-normalizable, so approximate constructions have to be considered.
- Row-Diagonal Parity (RDP) code[905] An MDS array code protecting against double erasures.
- Ruled-surface code[252,906] Evaluation code of polynomials evaluated on points lying on a ruled surface.
- SYK code[907,908]
- Sarvepalli-Brown subsystem code[435] Member of a family of non-CSS subsystem codes constructed from hypergraphs that satisfy certain assumptions [435; Construction C].
- Schubert code[913,914] Evaluation code of polynomials evaluated on points lying on a Schubert variety.
- Segre-variety RM-type code[915] Evaluation code of polynomials evaluated on points lying on a Segre variety.
- Self-complementary quantum code[916,917] A qubit code which admits a basis of codewords of the form \(|c\rangle+|\overline{c}\rangle\), where \(c\) is a bitstring and \(\overline{c}\) is its negation a.k.a. complement. Their codewords generalize the two-qubit Bell states and three-qubit GHZ states and are often called (qubit) cat states or poor-man's GHZ states. Such codes were originally pointed out to perform well against AD noise [917].
- Self-correcting quantum code[24,635] a.k.a. Self-correcting quantum memory, Thermally stable encoding.A block quantum code that forms the ground-state subspace of an \(n\)-body geometrically local Hamiltonian whose logical information is recoverable for arbitrary long times in the \(n\to\infty\) limit after interaction with a sufficiently cold thermal environment. Typically, one also requires a decoder whose decoding time scales polynomially with \(n\) and a finite energy density. The original criteria for a self-correcting quantum memory, informally known as the Caltech rules [355,918], also required finite-spin Hamiltonians.
- Self-dual additive code An additive \((n,2^n)_q\) code that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product, usually the trace-Hermitian inner product.
- Self-dual code over \(R\) An additive linear code \(C\) over a ring \(R\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product.
- Self-dual linear code An \([n,n/2]_q\) code that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to an inner product, most commonly either Euclidean or Hermitian. Self-dual codes exist only for even lengths and have dimension \(k=n/2\). A code that is equivalent to its dual is called iso-dual.
- Semakov-Zinoviev-Zaitsev (SZZ) equidistant code[919] Member of a family that is related to affine resolvable block designs and that is universally optimal.
- Sequential-recovery code[920,921] A \(t\)-erasure LRC whose coordinate erasures are recovered in sequential fashion.
- Sharp configuration[150,226,922] a.k.a. Delsarte code.A code \(C\) that attains a universal bound expressed in terms of the minimal distance, the number of distances between codewords, and the strength of the design formed by the codewords. For codes on a compact connected two-point homogeneous space, \(C\) is a design of strength \(M\) and admits \(m\) different distances between its points such that \(M \geq 2m - 1 - \delta\), where \(\delta\) is one if there are two antipodal points in \(C\) and zero otherwise [150].
- Sierpinsky fractal spin-liquid (SFSL) code[168,923] A fractal type-I fracton CSS code defined on a cubic lattice [19; Eq. (D22)]. The code admits an excitation-moving operator shaped like a Sierpinski triangle [19; Fig. 2].
- Simplex spherical code Spherical \((n,n+1,2+2/n)\) code whose codewords are all permutations of the \(n+1\)-dimensional vector \((1,1,\cdots,1,-n)\), up to normalization, forming an \(n\)-simplex. Codewords are all equidistant and their components add up to zero. Simplex spherical codewords in 2 (3, 4) dimensions form the vertices of a triangle (tetrahedron, 5-cell) In general, the code makes up the vertices of an \(n\)-simplex. See [116; Sec. 7.7] for a parameterization.
- Single parity-check (SPC) code a.k.a. Sum-zero code, Zero-sum code, Even-weight code.An \([n,n-1,2]\) linear binary code whose codewords consist of the message string appended with a parity-check bit or parity bit such that the parity (i.e., sum over all coordinates of each codeword) is zero. If the Hamming weight of a message is odd (even), then the parity bit is one (zero). This code requires only one extra bit of overhead and is therefore inexpensive. Its codewords are all even-weight binary strings. Its automorphism group is \(S_n\).
- Single-mode bosonic code
- Single-shot code[24,926,927] Block quantum qudit code whose error-syndrome weights increase linearly with the distance of the error state to the code space.
- Single-spin code An encoding into a monolithic (i.e. non-tensor-product) Hilbert space that houses an irreducible representation of \(SU(2)\) or, more generally, another Lie group. In some cases, this space can be thought of as the permutation invariant subspace of a particular tensor-product space.
- Singleton-bound approaching AQECC[351] Approximate quantum code of rate \(R\) that can tolerate adversarial errors nearly saturating the quantum Singleton bound of \((1-R)/2\). The formulation of such codes relies on a notion of quantum list decoding [351,928]. Sampling a description of this code can be done with an efficient randomized algorithm with \(2^{-\Omega(n)}\) failure probability.
- Six-qubit-tensor holographic code[514] Holographic tensor-network code constructed out of a network of encoding isometries of the \([[6,1,3]]\) six-qubit stabilizer code. The structure of the isometry is similar to that of the heptagon holographic code since both isometries are rank-six tensors, but the isometry in this case is neither a perfect tensor nor a planar-perfect tensor.
- Skew-cyclic CSS code[929,930] A member of a family of Galois-qudit CSS codes constructed from skew-cyclic classical codes over rings [930; Thm. 5.8]. See related study [931] that uses cyclic codes over rings.
- Skew-cyclic code[932] A classical code \(C\) of length \(n\) over an alphabet \(R\) is skew-cyclic if there exists an automorphism, \(\theta\), of \(R\), such that for each string \(c_1 c_2 \cdots c_n\in C\), the skew-cyclically shifted string \(\theta(c_n) \theta(c_1) \cdots \theta(c_{n-1})\in C\). We say that \(C\) is a \(\theta\)-cyclic code over \(R\).
- Slepian group-orbit code[620,933,934] Spherical code in \(n\) dimensions whose codewords correspond to points in an orbit of some initial vector under a generating group \(G\), which is a subgroup of the orthogonal group \(O(n)\) of rotations in \(n\) dimensions, i.e., the automorphism group of spherical codes under the Euclidean distance. Neither the vector nor the group are unique for a given code.
- Sloane-Whitehead code[935] Member of an infinite \((n,\lambda\cdot 2^{n-m-1},3)\) nonlinear code family for any \(n\) satisfying \(2^m \leq n < 3.2^{m-1}\) for some \(m\) and for \(\lambda\in\{20/16,19/16,18/16\}\). Such a code has more codewords than any linear code with the same length and distance. The code is constructed by applying the \((u|u+v)\) construction recursively to the Julin-Golay codes.
- Small-distance block code A block code of length \(n\) that either detects or corrects errors on at most two coordinates, i.e., has distance \(d \leq 5\).
- Small-distance block quantum code A block quantum code on \(n\) subsystems that either detects or corrects errors on at most two subsystems, i.e., have distance \(\leq 5\).
- Smith \(40\)-point code[936,937]
- Smolin-Smith-Wehner (SSW) code[916,938] A family of \(((n=4k+2l+3,M_{k,l},2))\) self-complementary CWS codes, where \(M_{k,l} \approx 2^{n-2}(1-\sqrt{2/(\pi(n-1))})\). For \(n \geq 11\), these codes have a logical subspace whose dimension is larger than that of the largest stabilizer code for the same \(n\) and \(d\). A subset of these codes can be augmented to yield codes with one higher logical dimension [939].
- Snub-cube code Spherical \((3,24,0.55384)\) code whose codewords are the vertices of the snub cube.
- Spacetime block code (STBC)[687,940–942] In a space-time block code, \(n\) spatially separated channels transmit symbols to \(m\) receiving channel using \(T\) time slots. These symbols can be arranged in a \(n \times T\) matrix where the rows correspond to the channels, and the columns correspond to the time slots. The codewords \(\{X\}\) making up the code are thus \(n \times T\) matrices.
- Spacetime circuit code[943–945] Qubit stabilizer code used to correct faults in Clifford circuits, i.e., circuits up made of Clifford gates and Pauli measurements. The code utilizes redundancy in the measurement outcomes of a circuit to correct circuit faults, which correspond to Pauli errors of the code.
- Spacetime code (STC)[946] Code designed for wireless transmission of information (via, e.g., radio waves) such that the sender can send multiple times from multiple locations. A spacetime code uses a modulation scheme to encode a message into signals that are sent at different times through different antennas, thereby utilizing both spatial and temporal (i.e., spacetime) degrees of freedom.
- Sparse subsystem code[943] A geometrically local qubit, modular-qudit, or Galois-qudit subsystem stabilizer code for which the number of sites participating in each gauge-group generator and the number of gauge-group generators that each site participates in are both bounded by a constant as \(n\to\infty\).
- Spatially coupled LDPC (SC-LDPC) code[598–600,947,948] LDPC code whose parity-check matrix is constructed by "spatially" coupling several copies of a regular LDPC parity-check matrix in chain-like fashion (or, more generally, in grid-like fashion) to yield a band matrix. A finite-length chain is then capped by imposing either open boundary conditions (yielding non-tail-biting SC-LDPC codes) or open boundary conditions (yielding tail-biting SC-LDPC codes); sometimes extra terminating vertices are added to the ends of the chain. Matrices corresponding to translationally invariant chains are called time-variant, and otherwise are called time-invariant. These codes can be constructed, e.g., using the lifting procedure or using edge-cutting vectors [949].
- Sphere packing Encodes states (codewords) into coordinates in the \(n\)-dimensional real coordinate space \(\mathbb{R}^n\). The number of codewords may be infinite because the coordinate space is infinite, so various restricted versions have to be constructed in practice.
- Spherical code Code whose codewords are points on an \(n\)-dimensional sphere \(S^{n}\) with radius one.
- Spherical design[950] Spherical code whose codewords are uniformly distributed in a way that is useful for determining averages of polynomials over the real sphere. A spherical code is a spherical design of strength \(t\), i.e., a \(t\)-design, if the average of any polynomial of degree up to \(t\) over its codewords is equal to the average over the entire sphere.
- Spherical sharp configuration[150,226,951,952] A spherical code that is a spherical design of strength \(2m-1\) for some \(m\) and that has \(m\) distances between distinct points. All known spherical sharp configrations are either obtained from the Leech or \(E_8\) lattice, certain regular polytopes, or are CGS isotropic subspace spherical codes [953; Table 1].
- Spin GKP code[954] An analogue of the single-mode GKP code designed for atomic ensembles. Was designed by using the Holstein-Primakoff mapping [955] (see also [956]) to pull back the phase-space structure of a bosonic system to the compact phase space of a quantum spin. A different construction emerges depending on which particular expression for GKP codewords is pulled back.
- Spin cat code[957,958] An analogue of the two-component cat code designed for a large spin, which is often realized in the PI subspace of atomic ensembles.
- Spin code Encodes \(K\)-dimensional Hilbert space into a tensor-product or direct sum of factors, with each factor spanned by states of a quantum mechanical spin or, more generally, an irreducible representation of a Lie group.
- Square-antiprism code Spherical \((3,8,4(4-\sqrt{2})/7)\) code whose codewords are the vertices of the square antiprism.
- Square-lattice GKP code[452] Single-mode GKP qudit-into-oscillator code based on the rectangular lattice. Its stabilizer generators are oscillator displacement operators \(\hat{S}_q(2\alpha)=e^{-2i\alpha \hat{p}}\) and \(\hat{S}_p(2\beta)=e^{2i\beta \hat{x}}\). To ensure \(\hat{S}_q(2\alpha)\) and \(\hat{S}_p(2\beta)\) generate a stabilizer group that is Abelian, there is a constraint that \(\alpha\beta=2q\pi\) where \(q\) is an integer denoting the logical dimension.
- Square-octagon (4.8.8) color code[7] Triangular color code defined on a patch of the 4.8.8 (square-octagon) tiling, which itself is obtained by applying a fattening procedure to the square lattice [8].
- Squeezed cat code[959–961] Two-component cat code whose two coherent states have been squeezed in a direction perpendicular to the segment formed by the two coherent state values \(\pm\alpha\).
- Squeezed fock-state code[962] Approximate bosonic code that encodes a qubit into the same Fock state, but one which is squeezed in opposite directions.
- Srivastava code[104,428] A special case of a generalized Srivastava code for \(z_j = \alpha_j^{\mu}\) for some \(\mu\) and \(t=1\).
- Stabilizer code A code whose logical subspace is the joint eigenspace (usually with eigenvalue \(+1\)) of a set of commuting unitary Pauli-type operators forming the code's stabilizer group. They can be block codes defined of tensor-product spaces of qubits or qudits, or non-block codes defined on single sufficiently large Hilbert spaces such as bosonic modes or group spaces.
- Star code[963] An MDS array code protecting against triple erasures.
- Stellated color code[964] A non-CSS color code on a lattice patch with a single twist defect at the center of the patch.
- String-net code[285,336,965,966] a.k.a. Levin-Wen model code, Turaev-Viro code.Code whose codewords realize a 2D topological order rendered by a Turaev-Viro topological field theory. The corresponding anyon theory is defined by a (multiplicity-free) unitary fusion category \( \mathcal{C} \). The code is defined on a cell decomposition dual to a triangulation of a two-dimensional surface, with a qudit of dimension \( |\mathcal{C}| \) located at each edge of the decomposition. These models realize local topological order (LTO) [967].
- Subspace code[968] A code that is a set of subspaces of \(GF(q)^n\).
- Subspace design[969,970] a.k.a. \(q\)-design, Geometric design.A \(q\)-ary code that can be mapped into a subspace \(t\)-\((n,w,\lambda)_q\) design.
- Subsystem CSS code[971–973] Subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) Pauli strings. This ensures that the code's stabilizer group is also CSS.
- Subsystem Galois-qudit CSS code[972,973] a.k.a. Euclidean construction subsystem code.Galois-qudit subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) Galois-qudit Pauli strings.
- Subsystem Galois-qudit code a.k.a. Gauge Galois-qudit code.Subsystem QECC encoding into a \(q^n\)-dimensional Hilbert space consisting of \(n\) Galois qudits.
- Subsystem Galois-qudit stabilizer code[971] a.k.a. Gauge Galois-qudit stabilizer code.Galois-qudit generalization of a subsystem qubit stabilizer code. Can be obtained by taking a Galois-qudit stabilizer code and assigning some of its logical qubits to be gauge qubits.
- Subsystem QECC[974,975] a.k.a. Operator QECC (OQECC), Gauge QECC.A quantum code which encodes quantum information in a tensor factor of a subspace that is decomposed into a tensor product of subsystems.
- Subsystem color code[12,22] a.k.a. Gauge color code.A subsystem version of the color code. One way to obtain it is by expanding the vertices of a two-colex embedded in a surface of genus \(g\). Vertex expansion consists of spl every vertex into a triangle and splitting every edge into a pair of edges.
- Subsystem homological product code[976] A CSS subsystem code constructed from a product of two (subspace) CSS codes. The case for qubits is formulated below, but these codes have also been extended to Galois qudits [976].
- Subsystem hyperbolic surface code[977] Subsystem generalization of the surface code on a 2D hyperbolic tesselation with gauge-group generators of weight at most three. An \(\{r,s\}\) hyperbolic tesselation with \(E\) edges yields a \([[3E/2,(1/2-2/r)E+2,(1-2/r)E,d]]\) subsystem code.
- Subsystem hypergraph product (SHP) code[427,978] a.k.a. Subsystem generalized Shor code, Bacon-Casaccino subsystem code.A CSS subsystem version of the generalized Shor code that has the same parameters as the subspace version, but requires fewer stabilizer measurements, resulting in a simpler error recovery routine. The code can also be thought of as a subsystem version of an HGP code because two such codes reduce to an HGP code upon gauge fixing [978; Sec. III]. The code can be obtained from a generalized Shor code by removing certain stabilizers that do no affect the code distance.
- Subsystem lifted-product (SLP) code[979] Member of a family of subsystem CSS codes constructed from a subsystem hypergraph product of a lifted binary linear code.
- Subsystem modular-qudit CSS code Modular-qudit subsystem stabilizer code which admits a set of gauge-group generators which consist of either all-\(Z\) or all-\(X\) modular-qudit Pauli strings. This ensures that the code's stabilizer group is also CSS.
- Subsystem modular-qudit code a.k.a. Gauge modular-qudit code.Subsystem QECC encoding into a \(q^n\)-dimensional Hilbert space consisting of \(n\) modular qudits.
- Subsystem modular-qudit stabilizer code a.k.a. Gauge modular-qudit stabilizer code.Modular-qudit generalization of a subsystem qubit stabilizer code. Can be obtained by taking a modular qudit stabilizer code and assigning some of its logical qubits to be gauge qubits. For composite qudit dimensions, such codes need not encode an integer number of qudits.
- Subsystem qubit code a.k.a. Gauge qubit code.Subsystem QECC encoding into a \(2^n\)-dimensional (i.e., \(n\)-qubit) Hilbert space.
- Subsystem qubit stabilizer code[980] a.k.a. Gauge qubit stabilizer code.A stabilizer code with some of its logical qubits denoted as gauge qubits and not used for storage of logical information. Note that this doesn't lead to new codes but does lead to new error correction and fault tolerance procedures. Subsystem codes are denoted by \([[n,k,g,d]]\), similar to stabilizer codes, but with an extra parameter \(g\) denoting the number of gauge qubits.
- Subsystem rotated surface code[981] Subsystem version of the rotated surface code.
- Subsystem spacetime circuit code[943,944] Subsystem stabilizer code obtained from a spacetime circuit code by gauging out logical operators that correspond to circuit faults with trivial effect [945; Sec. 5.4].
- Subsystem surface code[982] Subsystem version of the surface code defined on a square lattice with qubits placed at every vertex and center of everry edge.
- Sum-rank-metric code[983] A code whose performance is evaluated in the sum-rank metric, which is a metric that generalizes both the Hamming metric and the rank metric.
- Superimposed code[984–987] A set of binary strings such that taking a bitwise OR (e.g., \(1+1=1\)) of a small set of codewords does not yield another codeword.
- Surface-17 code[902] A \([[9,1,3]]\) rotated surface code named for the sum of its 9 data qubits and 8 syndrome qubits. It uses the smallest number of qubits to perform fault-tolerant error correction on a surface code with parallel syndrome extraction.
- Surface-code-fragment (SCF) holographic code[988] Holographic tensor-network code constructed out of a network of encoding isometries of the \([[5,1,2]]\) rotated surface code. The structure of the isometry is similar to that of the HaPPY code since both isometries are rank-six tensors. In the case of the SCF holographic code, the isometry is only a planar-perfect tensor (as opposed to a perfect tensor).
- Suzuki-curve code[989] Evaluation AG code of rational functions evaluated on points lying on a Suzuki curve.
- Symmetry-protected self-correcting quantum code[990] a.k.a. Symmetry-protected self-correcting memory.A code which admits a restricted notion of thermal stability against symmetric perturbations, i.e., perturbations that commute with a set of operators forming a group \(G\) called the symmetry group.
- Symmetry-protected topological (SPT) code[991,992] A code whose codewords form the ground-state or low-energy subspace of a code Hamiltonian realizing symmetry-protected topological (SPT) order.
- Ta-Shma zigzag code[993] Member of a family of \(\epsilon\)-balanced codes that nearly achieves the asymptotic GV bound. The codes have relative distance \(\frac{1}{2}-\frac{\epsilon}{2}\) and rate of order \(\Omega (\epsilon^{2+\beta})\) for \(\beta\to 0\) as \(n\to\infty\) [994].
- Tamo-Barg code[995] A family of \(q\)-ary polynomial evaluation codes that are optimal LRCs and for which \(q\) is comparable to \(n\).
- Tamo-Barg-Vladut code[996,997] Polynomial evaluation code on algebraic curves, such as Hermitian or Garcia-Stichtenoth curves, that is constructed to be an LRC. Codes can be constructed to be be able to recover locally after one or more erasures as well as to tackle the availability problem.
- Tanner code[892] a.k.a. Generalized LDPC (GLDPC) code.A linear \(q\)-ary code defined on a bipartite graph similar to a Tanner graph. This generalized Tanner graph consists of variable nodes and constraint nodes, with the generalization being that the constraint nodes of degree \(r\) now represent a linear codes of length \(r\).
- Tanner-Sridhara-Fuja (TSF) code[48] Array QC-LDPC code constructed from a cyclically shifted identity matrix; see [998; Exam. 21.6.5].
- Tensor-network code[137,514,999–1001] a.k.a. Quantum Lego code.Block quantum code constructed using a tensor-network-based graphical framework from atomic tensors a.k.a. quantum Lego blocks [999], which can be encoding isometries for smaller quantum codes. The class of codes constructed using the framework depends on the choice of atomic Lego blocks.
- Tensor-product HDX code[1002] Code constructed in a similar way as the HDX code, but utilizing tensor products of multiple Ramanujan complexes and then applying distance balancing. These improve the asymptotic code distance over the HDX codes from \(\sqrt{n}\log n\) to \(\sqrt{n}~\text{polylog}(n)\). The utility of such tensor products comes from the fact that one of the Ramanujan complexes is a collective cosystolic expander as opposed to just a cosystolic expander.
- Tensor-product code[214,1003–1005] a.k.a. Tensor code, Kroneckerian code, Product code.A matrix-based code constructed out of two linear binary or \(q\)-ary codes \(C_A,C_B\) in an outer-product construction denoted as \(C_A \otimes C_B\). Its dual is sometimes called the check-product code, denoted as \(C_{A}\boxplus C_{B}\).
- Ternary Golay code[441] A \([11,6,5]_3\) perfect ternary linear code with connections to various areas of mathematics, e.g., lattices [110] and sporadic simple groups [56]. Adding a parity bit to the code results in the self-dual \([12,6,6]_3\) extended ternary Golay code. Up to equivalence, both codes are unique for their respective parameters [442]. The dual of the ternary Golay code is a \([11,5,6]_3\) projective two-weight subcode.
- Ternary-tree fermion-into-qubit code[1006] A fermion-into-qubit encoding defined on ternary trees that maps Majorana operators into Pauli strings of weight \(\lceil \log_3 (2n+1) \rceil\).
- Tetracode[110] The \([4,2,3]_3\) self-dual MDS code that has connections to lattices [110].
- Tetrahedral color code[22,1007] 3D color code defined on select tetrahedra of a 3D tiling. Qubits are placed on the vertices, edges, triangles, and in the center of each tetrahedron. The code has both string-like and sheet-like logical operators [1008].
- Three-fermion (3F) Walker-Wang model code[1009,1010]
- Three-fermion (3F) subsystem code[165,1012] 2D subsystem stabilizer code whose low-energy excitations realize the three-fermion anyon theory [1011–1013]. One version uses two qubits at each site [165], while other manifestations utilize a single qubit per site and only weight-two (two-body) interactions [12,1012]. All are expected to be equivalent to each other via a local constant-depth Clifford circuit.
- Three-qutrit code[1014] A \([[3,1,2]]_3\) prime-qudit CSS code that is the smallest qutrit stabilizer code to detect a single-qutrit error. with stabilizer generators \(ZZZ\) and \(XXX\). The code defines a quantum secret-sharing scheme and serves as a minimal model for the AdS/CFT holographic duality. It is also the smallest non-trivial instance of a quantum maximum distance separable code (QMDS), saturating the quantum Singleton bound.
- Three-rotor code[230] \([[3,1,2]]_{\mathbb Z}\) rotor code that is an extension of the \([[3,1,2]]_3\) qutrit CSS code to the integer alphabet, i.e., the angular momentum states of a planar rotor.
- Topological code[35] A code whose codewords form the ground-state or low-energy subspace of a (typically geometrically local) code Hamiltonian realizing a topological phase. A topological phase may be bosonic or fermionic, i.e., constructed out of underlying subsystems whose operators commute or anti-commute with each other, respectively. Unless otherwise noted, the phases discussed are bosonic.
- Toric code[75,593] Version of the Kitaev surface code on the two-dimensional torus, encoding two logical qubits. Being the first manifestation of the surface code, "toric code" is often an alternative name for the general construction. Twisted toric code [94; Fig. 8] refers to the construction on a torus with twisted (a.k.a. shifted) boundary conditions.
- Tornado code[352,565,1015] Linear binary code that is a precursor to fountain codes and whose encoding and decoding operations involve only XOR gates [1016; Sec. 30.2].
- Torus-layer spherical code (TLSC)[1017] Code whose codewords are elements of a foliation of the \(2n-1\)-dimensional hypersphere \(S^{2n-1}\) using flat tori \(S^1\times S^1\cdots\times S^1\). Related constructions include the spherical codes by Hopf foliations (SCHF) [1018].
- Traceability code[1019] An IPP code with which it is possible to detect a parent of a given pirated descendent by finding the closest codeword to that descendant.
- Transverse-field Ising model (TFIM) code[1020] A 1D translationally invariant stabilizer code whose encoding is a constant-depth circuit of nearest-neighbor gates on alternating even and odd bonds that consist of transverse-field Ising Hamiltonian interactions. The code allows for perfect state transfer of arbitrary distance using local operations and classical communications (LOCC).
- Tree cluster-state code[1021–1023] Code obtained from a cluster state on a tree graph that has been proposed in the context of quantum repeater and MBQC architectures.
- Triangular surface code[1024] a.k.a. Triangle surface code.A surface code with weight-four stabilizer generators defined on a triangular lattice patch that are examples of twist-defect surface code with a single twist defect at the center of the patch. The codes use about \(25\%\) fewer physical per logical qubit for a given distance compared to the surface code.
- Triorthogonal code[1025] Qubit CSS code whose \(X\)-type logicals and stabilizer generators form a triorthogonal matrix (defined below) in the symplectic representation.
- True Galois-qudit stabilizer code[216,415,416] a.k.a. Linear stabilizer code.A \([[n,k,d]]_q\) stabilizer code whose stabilizer's Galois symplectic representation forms a linear subspace. In other words, the set of \(q\)-ary vectors representing the stabilizer group is closed under both addition and multiplication by elements of \(GF(q)\). In contrast, Galois-qudit stabilizer codes admit sets of vectors that are closed under addition only.
- Truncated trihexagonal (4.6.12) color code[1026] Triangular color code defined on a patch of the 4.6.12 (truncated trihexagonal or square-hexagon-dodecagon) tiling.
- Tsfasman-Vladut-Zink (TVZ) code[1027] Member of a family of residue AG or, more generally, evaluation AG codes where \(\cal X\) is either Drinfeld modular curve, a classic modular curve, or a Garcia-Stichtenoth curve.
- Turbo code[1028,1029] Code obtained from a parallel concatenation of two or more convolutional codes with permutations interleaving the individual encodings.
- Twist-defect color code[899,1030,1031] a.k.a. Color code with a twist.A non-CSS extension of the 2D color code whose non-CSS stabilizer generators are associated with twist defects of the associated lattice.
- Twist-defect surface code[899,964,1024,1032–1035] a.k.a. Surface code with a twist, Genon surface code.A non-CSS extension of the 2D surface-code construction whose non-CSS stabilizer generators are associated with twist defects of the associated lattice. A related construction [1035] doubles the number of qubits in the lattice via symplectic doubling.
- Twisted BCH code[1036–1038] a.k.a. RS subspace subcode.Additive or linear \(q\)-ary code obtained from BCH codes via various lengthening and extension procedures such as Construction X and Construction XX.
- Twisted XZZX toric code[1039] a.k.a. XZZX cyclic code, Cyclic toric code, Generalized toric code (GTC).A cyclic code that can be thought of as the XZZX toric code with shifted (a.k.a twisted) boundary conditions. Admits a set of stabilizer generators that are cyclic shifts of a particular weight-four \(XZZX\) Pauli string. For example, a seven-qubit \([[7,1,3]]\) variant has stabilizers generated by cyclic shifts of \(XZIZXII\) [1040]. Codes encode either one or two logical qubits, depending on qubit geometry, and perform well against biased noise [1041].
- Twisted \(1\)-group code[1042,1043] Block group-representation code realizing particular irreps of particular groups such that a distance of two is automatically guaranteed. Groups which admit irreps with this property are called twisted (unitary) \(1\)-groups and include the binary icosahedral group \(2I\), the \(\Sigma(360\phi)\) subgroup of \(SU(3)\), the family \(\{PSp(2b, 3), b \geq 1\}\), and the alternating groups \(A_{5,6}\). Groups whose irreps are images of the appropriate irreps of twisted \(1\)-groups also yield such properties, e.g., the binary tetrahedral group \(2T\) or qutrit Pauli group \(\Sigma(72\phi)\).
- Twisted quantum double (TQD) code[31,32,1044] Code whose codewords realize a 2D topological order rendered by a Chern-Simons topological field theory. The corresponding anyon theory is defined by a finite group \(G\) and a Type-III group cocycle \(\omega\), but can also be described in a category theoretic way [1045].
- Two-block CSS code[429] a.k.a. Two-sublattice code, Two-square-block code.Galois-qudit CSS code whose stabilizer generator matrices \(H_X=(A,B)\) and \(H_Z=(B^T,-A^T)\), are constructed from a pair of square commuting matrices \(A\) and \(B\).
- Two-block group-algebra (2BGA) codes[1046–1048] a.k.a. Non-Abelian GB code, LR code.2BGA codes are the smallest LP codes LP\((a,b)\), constructed from a pair of group algebra elements \(a,b\in \mathbb{F}_q[G]\), where \(G\) is a finite group, and \(\mathbb{F}_q\) is a Galois field. For a group of order \(\ell\), we get a 2BGA code of length \(n=2\ell\). A 2BGA code for an Abelian group is called an Abelian 2BGA code. A construction of such codes in terms of Kronecker products of circulant matrices was introduced in [429].
- Two-component cat code[1049] Code whose codespace is spanned by two coherent states \(\left|\pm\alpha\right\rangle\) for nonzero complex \(\alpha\).
- Two-gauge theory code[1050] a.k.a. Higher gauge theory code.A code whose codewords realize lattice two-gauge theory [1051–1056] for a finite two-group (a.k.a. a crossed module) in arbitrary spatial dimension. There exist several lattice-model formulations in arbitrary spatial dimension [1050,1057] as well as explicitly in 3D [1058–1061] and 4D [1061], with the 3D case realizing the Yetter model [1062–1065].
- Two-mode binomial code[58] Two-mode constant-energy CLY code whose coefficients are square-roots of binomial coefficients.
- Two-weight code A linear \(q\)-ary code whose codewords all have one of two possible nonzero Hamming weights.
- Type-II fractal spin-liquid code[168] A type-II fracton prime-qudit CSS code defined on a cubic lattice [19; Eqs. (D9-D10)].
- Unary code a.k.a. Thermometer code.Trivial code that encodes integers \(1\) through \(n\) into binary strings of length \(n\). The \(i\)th codeword is a string consisting of \(i\) ones followed by \(n-i\) zeroes.
- Uniformly packed code[700,1066,1067] An \((n,K,2t+1)_q\) code is uniformly packed if its external distance is equal to \(t+1\) [56]; see [204; Def. 2.5] for the case of even distance and other generalizations.
- Unimodular lattice code a.k.a. Self-dual lattice code.A lattice that is equal to its dual, \(L^\perp = L\). Unimodular lattices have \(\det L = \pm 1\).
- Union stabilizer (USt) code[407–411] a.k.a. Non-stabilizer code, Quotient space quantum code (QSQC).A qubit code whose codespace consists of a direct sum of a qubit stabilizer codespace and one or more of that stabilizer code's error spaces.
- Union-Jack color code[1068] Triangular color code defined on a patch of the Tetrakis square tiling (a.k.a. the Union Jack lattice).
- Universally optimal \(q\)-ary code[226,922,1069–1073] A binary or \(q\)-ary code that (weakly) minimizes all completely monotonic potentials on binary space [1073].
- Universally optimal code[1074] A code that produces a minimum over all codes of its cardinality for a large family of potential functions. Such codes exist for the conventional \(q\)-ary and real spaces (see children below), but can also be formulated for more exotic spaces such as Lie groups, projective spaces, and real Grassmanians [1075,1076].
- Universally optimal sphere packing[150] A periodic sphere packing that (weakly) minimizes all completely monotonic potentials of square Euclidean distance among all periodic packings of the same density.
- Universally optimal spherical code[150,1072,1077–1079] A spherical code that (weakly) minimizes all completely monotonic potentials on the sphere for its cardinality. See [1081][1080; Sec. 12.4] for further discussion.
- Valence-bond-solid (VBS) code[1082,1083] An \(n\)-qubit approximate \(q\)-dimensional spin code family whose codespace is described in terms of \(SU(q)\) valence-bond-solid (VBS) [1084] matrix product states with various boundary conditions. The codes become exact when either \(n\) or \(q\) go to infinity.
- Varshamov-Tenengolts (VT) code[1085,1086] Nearly optimal binary deletion-correcting code and code for the asymmetric channel.
- Vasilyev code[1087] Member of an infinite \((2^{m+1}-1,2^{2n-m},3)\) family of perfect nonlinear codes for any \(m \geq 3\). Constructed by applying a modification of the \((u|u+v)\) construction to a perfect \((2^m-1,2^{n-m},3)\) code, not necessarily linear [56; pg. 77].
- Very small logical qubit (VSLQ) code[1088,1089] The two logical codewords are \(|\pm\rangle \propto (|0\rangle\pm|2\rangle)(|0\rangle\pm|2\rangle)\), where the total Hilbert space is the tensor product of two transmon qudits (whose ground states \(|0\rangle\) and second excited states \(|2\rangle\) are used in the codewords). Since the code is intended to protect against losses, the qutrits can equivalently be thought of as oscillator Fock-state subspaces.
- W-state code[345] Approximate block quantum code whose encoding resembles the structure of the W state [1090]. This code enables universal quantum computation with transversal gates.
- Walker-Wang model code[16] A 3D topological code defined by a unitary braided fusion category \( \mathcal{C} \) (also known as a unitary premodular category). The code is defined on a cubic lattice that is resolved to be trivalent, with a qudit of dimension \( |\mathcal{C}| \) located at each edge. The codespace is the ground-state subspace of the Walker-Wang model Hamiltonian [16] and realizes the Crane-Yetter model [1091–1093]. A single-state version of the code provides a resource state for MBQC [1010].
- Wasilewski-Banaszek code[1094] Three-oscillator constant-excitation Fock-state code encoding a single logical qubit.
- Weighed-covering code A \(q\)-ary code for which balls of some radius centered at its codewords provide a weighted covering of the Hamming space.
- Weight-two code[1095] A length-\(n\) binary code whose codewords all have Hamming weight two. Such codes provide slightly extra redundancy for storage of small-scale information such as ZIP codes or decimal digits.
- Witting polytope code a.k.a. \(4_{21}\) real polytope code.Spherical \((8,240,1)\) code whose codewords are the vertices of the Witting complex polytope, the \(4_{21}\) real polytope, and the minimal lattice-shell code of the \(E_8\) lattice. The code is optimal and unique up to equivalence [110,1096,1097]. Antipodal pairs of points correspond to the 120 tritangent planes of a canonic sextic curve [150,496–498].
- Wozencraft ensemble code[1098] A code that is part of the Wozencraft ensemble, a set of codes most of whose members achieve the GV bound.
- Wrapped spherical code[1099] Spherical code in dimension \(n\) whose codewords are obtained from centers of spheres from a finite \(S^{n-1}\)-sphere packing of \(\mathbb{R}^{n}\) that is "wrapped" onto \(S^n\).
- X-code[1100] An MDS array code with a simple geometrical construction that achieves optimal encoding and update complexity.
- X-cube Floquet code[1101] Floquet code whose qubits are placed on vertices of a truncated cubic lattice. Its weight-two check operators are placed on various edges. Its ISG can be that of the X-cube model code or that of several decoupled surface codes.
- X-cube model code[1102]
- XP stabilizer code[1106] a.k.a. Weighed hypergraph code.The XP Stabilizer formalism is a generalization of the XS and Pauli stabilizer formalisms, with stabilizer generators taken from the group \( \mathsf{BD}_{2N}^{\otimes n} = \langle\omega I, X, P\rangle^{\otimes n} \), which is the tensor product of the binary dihedral group of order \(8N\). Here, \(N\) is called the precision, \( \omega \) is a \( 2N \)th root of unity, and \( P = \text{diag} ( 1, \omega^2) \). The codespace is a \(+1\) eigenspace of a set of XP stabilizer generators, which need not commute to define a valid codespace.
- XS stabilizer code[1107] A type of stabilizer code where stabilizer generators are elements of the group \( \{\alpha I, X, \sqrt{Z}]\}^{\otimes n} \), with \( \sqrt{Z} = \text{diag} (1, i)\). The codespace is a joint \(+1\) eigenspace of a set of stabilizer generators, which need not commute to define a valid codespace.
- XY surface code[1108] a.k.a. Tailored surface code (TSC).Non-CSS derivative of the surface code whose generators are \(XXXX\) and \(YYYY\), obtained by mapping \(Z \to Y\) in the surface code.
- XYZ color code[1109] Non-CSS variant of the 6.6.6 color code whose generators are \(XZXZXZ\) and \(ZYZYZY\) Pauli strings associated to each hexagonal in the hexagonal (6.6.6) tiling.
- XYZ product code[1110,1111] A non-CSS QLDPC code constructed from a particular hypergraph product of three classical codes. The idea is that rather than taking a product of only two classical codes to produce a CSS code, a third classical code is considered, acting with Pauli-\(Y\) operators. When the underlying classical codes are 1D (e.g., repetition codes), the XYZ product yields a 3D code. Higher dimensional versions have been constructed [1112].
- XYZ ruby Floquet code[1113] Floquet code whose qubits are placed on vertices of a ruby lattice. Its weight-two check operators are placed on various edges. One third of the time during its measurement schedule, its ISG is that of the 6.6.6 color code concatenated with a three-qubit repetition code. Together, all ISGs generate the gauge group of the 3F subsystem code. A Floquet code with the same underlying subsystem code but with a different measurement schedule was developed in Ref. [1114].
- XYZ\(^2\) hexagonal stabilizer code[1115,1116] An instance of the matching code based on the Kitaev honeycomb model. It is described on a hexagonal lattice with \(XYZXYZ\) stabilizers on each hexagonal plaquette. Each vertical pair of qubits has an \(XX\), \(YY\), or \(ZZ\) link stabilizer depending on the orientation of the plaquette stabilizers.
- XZZX surface code[1039,1117–1119] a.k.a. Wen plaquette model.Non-CSS variant of the rotated surface code whose generators are \(XZZX\) Pauli strings associated, clock-wise, to the vertices of each face of a two-dimensional lattice (with a qubit located at each vertex of the tessellation).
- Ye-Barg code[1120,1121] An MDS array code with the optimal access property; see Ref. [1120] for definitions.
- Yoked surface code[843] Member of a family of \([[n,k,d]]\) qubit CSS codes resulting from a concatenation of a QMDPC code with a rotated surface code. Concatenation does not impose additional connectivity constraints and can as much as triple the number of logical qubits per physical qubit when compared to the original surface code. Concatenation with 1D (2D) QMDPC yields codes with twice (four times) the distance. The stabilizer generators of the outer QMDPC code are referred to as yokes in this context.
- Zero-pi qubit code[520,1122,1123]
- Zetterberg code[1124] Family of binary cyclic \([2^{2s}+1,2^{2s}-4s+1]\) codes with distance \(d>5\) generated by the minimal polynomial \(g_s(x)\) of \(\alpha\) over \(GF(2)\), where \(\alpha\) is a primitive \(n\)th root of unity in the field \(GF(2^{4s})\). They are quasi-perfect codes and are one of the best known families of double-error correcting binary linear codes
- Zigzag code[1125] An MDS array code correcting against two erasures with optimal rebuilding ratio; see Ref. [1125] for definitions.
- \(((10,24,3))\) qubit code[939] Ten-qubit CWS code that is unique and optimal for its parameters.
- \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code[411,1126] Member of a family of \(((2^m,2^{2^m−5m+1},8))\) CSS-like union stabilizer codes constructed using the classical Goethals and Preparata codes.
- \(((3,6,2))_{\mathbb{Z}_6}\) Euler code[1127] Six-qudit error-detecting code with logical dimension \(K=6\) that is obtained from a particular AME state that serves as a solution to the 36 officers of Euler problem. The code is obtained from a \(((4,1,3))_{\mathbb{Z}_6}\) code.
- \(((5+2r,3\times 2^{2r+1},2))\) Rains code[407] Member of a family of \(((5+2r,3\times 2^{2r+1},2))\) CWS codes; see also [938,1128][186; Exam. 8].
- \(((5,3,2))_3\) qutrit code[1042] Smallest qutrit block code realizing the \(\Sigma(360\phi)\) subgroup of \(SU(3)\) transversally. The next smallest code is \(((7,3,2))_3\).
- \(((5,6,2))\) qubit code[407] Six-qubit cyclic CWS code detecting a single-qubit error. This code has a logical subspace whose dimension is larger than that of the \([[5,2,2]]\) code, the best five-qubit stabilizer code with the same distance [939].
- \(((7,2))\) QETC[836] Seven-qubit QETC that transmutes all single-qubit Pauli errors to logical phase errors. See [836; Table 1] for its stabilizer generators.
- \(((7,2,3))\) Pollatsek-Ruskai code[176,473,770] Seven-qubit PI code that realizes gates from the binary icosahedral group transversally. Can also be interpreted as a spin-\(7/2\) single-spin code. The codespace projection is a projection onto an irrep of the binary icosahedral group \(2I\).
- \(((9,12,3))\) qubit code[1129] Nine-qubit cyclic CWS code correcting a single-qubit error. This code has a logical subspace whose dimension is larger than that of the \([[9,3,3]]\) code, the best nine-qubit stabilizer code with the same distance [282].
- \(((9,2,3))\) Ruskai code[1130] Nine-qubit PI code that protects against single-qubit errors as well as two-qubit errors arising from exchange processes.
- \(((n,1+n(q-1),2))_q\) union stabilizer code[410] Member of a family of \(((n,1+n(q-1),2))_q\) Galois-qudit union stabilizer code for odd \(n\).
- \(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code[1131] PI distance-two code on \(n\geq4\) qubits whose degree of entanglement vanishes asymptotically with \(n\) [1131; Appx. D] (cf. [1132]).
- \((1,3)\) 4D toric code[1133] A generalization of the Kitaev surface code defined on a 4D lattice. The code is called a \((1,3)\) toric code because it admits 1D \(Z\)-type and 3D \(X\)-type logical operators.
- \((5,1,2)\)-convolutional code[822] Family of quantum convolutional codes that are 1D lattice generalizations of the five-qubit perfect code, with the former''s lattice-translation symmetry being the extension of the latter''s cyclic permutation symmetry.
- \((u|u+v)\)-construction code[935,1134] Code constructed using a concatenation procedure that takes in two \(q\)-ary codes \(C_1,C_2\) and produces a new code whose codewords are \((u|u+v)\) for all \(u\in C_1\) and \(v\in C_2\). If the two codes have parameters \((n,K_1,d_1)\) and \((n,K_2,d_2)\), then the output code is a \((2n,K_1 K_2, \min\{2d_1,d_2\})\) code [70; Thm. 5.10][56; pg. 76].
- \(3_{21}\) polytope code[494] a.k.a. Hess polytope code, Hesse polytope code, 7-ic semi-regular figure code.Spherical \((7,56,1/3)\) code whose codewords are the vertices of the \(3_{21}\) real polytope (a.k.a. the Hess polytope). The vertices form the kissing configuration of the Witting polytope code. The code is optimal and unique up to equivalence [110,1096,1097]. Antipodal pairs of points correspond to the 28 bitangent lines of a general quartic plane curve [150,496–498].
- \(A_2\) hexagonal lattice code Two-dimensional lattice that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. Its dual is the honeycomb lattice. The ruby lattice is a fattened honeycomb lattice interpolating between the honeycomb and hexagonal lattices.
- \(A_n\) lattice code Lattice-based \(n\)-dimensional code that can be simply defined in \(n+1\) dimensions as the set of integer vectors \(x\) lying in the hyperplane \(x_0+x_1+\cdots+x_{n} = 0\).
- \(A_n^{\perp}\) lattice code Lattice-based \(n\)-dimensional code whose codewords form the dual of the \(A_n\) lattice.
- \(BW_{32}\) Barnes-Wall lattice code[98] BW lattice in dimension \(32\).
- \(BW_{32}\) lattice-shell code Spherical code whose codewords are points on the \(BW_{32}\) Barnes-Wall lattice normalized to lie on the unit sphere.
- \(D\)-dimensional twisted toric code[556] Extenstion of the Kitaev toric code to higher-dimensional lattices with shifted (a.k.a twisted) boundary conditions. Such boundary conditions yields quibit geometries that are tori \(\mathbb{R}^D/\Lambda\), where \(\Lambda\) is an arbitrary \(D\)-dimensional lattice. Picking a hypercubic lattice yields the ordinary \(D\)-dimensional toric code. It is conjectured that appropriate twisted boundary conditions yield multi-dimensional toric code families with linear distance and logarithmic-weight stabilizer generators [556].
- \(D_3\) face-centered cubic (fcc) lattice code a.k.a. Cannonball lattice.Laminated three-dimensional lattice consisting of layers of hexagonal lattices.
- \(D_4\) hyper-diamond GKP code[1135] Two-mode GKP qudit-into-oscillator code based on the \(D_4\) hyper-diamond lattice.
- \(D_4\) hyper-diamond lattice code BW lattice in dimension \(4\), which is the lattice corresponding to the \([4,1,4]\) repetition and \([4,3,2]\) SPC codes via Construction A.
- \(D_4\) lattice-shell code Spherical code whose codewords are points on the \(D_4\) lattice normalized to lie on the unit sphere.
- \(D_n\) checkerboard lattice code Lattice code consisting of all points whose coordinates add up to an even integer.
- \(ED_m\) code[1136] a.k.a. Equidistant code with maximal distance.Member of the family of \( (c\frac{qt-1}{(t-1,q-1)},qt,ct\frac{q-1}{(t-1,q-1)}) \) \(q\)-ary codes, where \(c,t\geq 1\) and \((a,b)\) is the greatest common divisor of \(a\) and \(b\). Such codes are universally optimal and are related to resolvable block designs.
- \(E_6\) lattice-shell code Spherical code whose codewords are points on the \(E_6\) lattice normalized to lie on the unit sphere.
- \(E_6\) root lattice code Lattice in dimension \(6\).
- \(E_7\) lattice-shell code Spherical code whose codewords are points on the \(E_7\) lattice normalized to lie on the unit sphere.
- \(E_7\) root lattice code Lattice in dimension \(7\).
- \(E_8\) Gosset lattice code[494] Unimodular even BW lattice in dimension \(8\), consisting of the Cayley integral octonions rescaled by \(\sqrt{2}\). The lattice corresponds to the \([8,4,4]\) Hamming code via Construction A.
- \(E_8\) Gosset lattice-shell code Spherical code whose codewords are points on the \(E_8\) Gosset lattice normalized to lie on the unit sphere.
- \(G\)-covariant erasure code[230] A \(G\)-covariant block code that serves as a proof-of-principle construction to demonstrate the existence of \(G\)-covariant codes where \(G\) is a finite group, and the physical space is finite-dimensional. This construction can be done for any erasure-correcting code.
- \(G\)-enriched Walker-Wang model code[1137] a.k.a. Williamson-Wang model code.A 3D topological code defined by a unitary \(G\)-crossed braided fusion category \( \mathcal{C} \) [1138,1139], where \(G\) is a finite group. The model realizes TQFTs that include two-gauge theories, those behind Walker-Wang models, as well as the Kashaev TQFT [1140,1141]. It has been generalized to include domain walls [1142].
- \(R\)-linear code A code of length \(n\) over a ring \(R\) is \(R\)-linear if it is a submodule of \(R^n\).
- \(SU(3)\) spin code[1143] An extension of Clifford single-spin codes to the group \(SU(3)\), whose codespace is a projection onto a particular irrep of a subgroup of \(SU(3)\) of an underlying spin that houses some particular irrep of \(SU(3)\).
- \(U(d)\)-covariant approximate erasure code[1144,1145] Covariant code whose construction takes in an arbitrary erasure-correcting code to yield an approximate QECC that is also covariant with respect to the unitary group.
- \([2^m,m+1,2^{m-1}]\) First-order RM code a.k.a. Biorthogonal code, RM\((1,m)\) code, Augmented Hadamard code.A member of the family of first-order RM codes. Its codewords are the rows of the \(2^m\)-dimensional Hadamard matrix \(H\) and its negation \(-H\) with the mapping \(+1\to 0\) and \(-1\to 1\). They form a \((2^m,2^{m+1})\) biorthogonal spherical code under the antipodal mapping.
- \([2^m-1,m,2^{m-1}]\) simplex code[337,1146] a.k.a. Shortened Hadamard code, RM\(^*(1,m)\) code, Maximum-length feedback-shift-register code.A member of the code family that is dual to the \([2^m,2^m-m-1,3]\) Hamming family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((2^m,2^m+1)\) simplex spherical code under the antipodal mapping.
- \([2^r,2^r-r-1,4]\) Extended Hamming code[337,441,1147] Member of an infinite family of binary linear codes with parameters \([2^r,2^r-r-1, 4]\) for \(r \geq 2\) that are extensions of the Hamming codes by a parity-check bit.
- \([2^r-1,2^r-r-1,3]\) Hamming code[441,1147] a.k.a. RM\(^*(r-2,r)\) code.Member of an infinite family of perfect linear codes with parameters \([2^r-1,2^r-r-1, 3]\) for \(r \geq 2\). Their \(r \times (2^r-1) \) parity-check matrix \(H\) has all possible non-zero \(r\)-bit strings as its columns. Adding a parity check yields the \([2^r,2^r-r-1, 4]\) extended Hamming code.
- \([48,24,12]\) self-dual code An extended quadratic-residue code that is known to be the only self-dual doubly even code at its parameters [1148].
- \([56,6,36]_3\) Hill-cap code[1149] Projective two-weight ternary code based on the Games graph [1151][1150; Table 19.1] and Hill's 56-cap [1149]. Its automorphism group contains \(PSL_3(4)\) [1152].
- \([7,3,4]\) simplex code a.k.a. RM\(^*(1,3)\) code, Little Hamming code.Second-smallest member of the simplex code family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(2,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((8,9)\) simplex spherical code under the antipodal mapping.
- \([7,4,3]\) Hamming code[337,441,1147] Second-smallest member of the Hamming code family.
- \([78,6,56]_4\) Hill-cap code[1153] Projective two-weight quaternary code based on a cap that corresponds to a strongly regular graph [1151; Table 7.1].
- \([8,4,4]\) extended Hamming code[337,441,1147] Extension of the \([7,4,3]\) Hamming code by a parity-check bit. The smallest doubly even self-dual code.
- \([[10,1,2]]\) CSS code[95] Smallest stabilizer code to implement a logical \(T\) gate via application of physical \(T\), \(T^{\dagger}\), and \(CCZ\) gates.
- \([[10,1,4]]_{G}\) tenfold code[454; Prop. V.1] A \([[10,1,4]]_{G}\) group code for finite Abelian \(G\). The code is defined using a graph that is closely related to the \([[5,1,3]]\) code.
- \([[11,1,5]]\) quantum dodecacode[282] Eleven-qubit pure stabilizer code that is the smallest qubit stabilizer code to correct two-qubit errors.
- \([[11,1,5]]_3\) qutrit Golay code[1154] An \([[11,1,5]]_3\) constructed from the ternary Golay code via the CSS construction. The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix of the ternary Golay code.
- \([[12,2,4]]\) carbon code[1155] Self-dual twelve-qubit CSS code.
- \([[13,1,5]]\) cyclic code[1039] Thirteen-qubit twisted surface code for which there is a set of stabilizer generators consisting of cyclic permutations of the \(XZZX\)-type Pauli string \(XIZZIXIIIIIII\). The code can be thought of as a small twisted XZZX code [1039; Ex. 11 and Fig. 3].
- \([[144,12,12]]\) gross code[125] A BB QLDPC code which requires less physical and ancilla qubits (for syndrome extraction) than the surface code with the same number of logical qubits and distance. The name stems from the fact that a gross is a dozen dozen.
- \([[15, 7, 3]]\) quantum Hamming code[82,395,1156] Self-dual quantum Hamming code that admits permutation-based CZ logical gates. The code is constructed using the CSS construction from the \([15,11,3]\) Hamming code and its \([15,4,8]\) dual code.
- \([[15,1,3]]\) quantum Reed-Muller code[817,1157,1158] \([[15,1,3]]\) CSS code that is most easily thought of as a tetrahedral 3D color code.
- \([[16,6,4]]\) Tesseract color code[95,1159] A 4D color code defined on a tesseract, with stabilizer generators of both types supported on each cube.
- \([[2^D,D,2]]\) hypercube quantum code[1160,1161][95; Exam. 3] a.k.a. Hyperoctahedron code, Hyperoctahedron color code.Member of a family of codes defined by placing qubits on a \(D\)-dimensional hypercube, \(Z\)-stabilizers on all two-dimensional faces, and an \(X\)-stabilizer on all vertices. These codes realize gates at the \((D-1)\)-st level of the Clifford hierarchy. It can be generalized to a \([[4^D,D,4]]\) error-correcting family [1162]. Various other concatenations give families with increasing distance (see cousins).
- \([[2^r, 2^r-r-2, 3]]\) Gottesman code[1163] a.k.a. \([[2^r, 2^r-r-2, 3]]\) quantum Hamming code.A family of non-CSS stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound.
- \([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code[82] Member of a family of self-dual CCS codes constructed from \([2^r-1,2^r-r-1,3]=C_X=C_Z\) Hamming codes and their duals the simplex codes. The code's stabilizer generator matrix blocks \(H_{X}\) and \(H_{Z}\) are both the generator matrix for a simplex code. The weight of each stabilizer generator is \(2^{r-1}\).
- \([[2^r-1, 2^r-2r-1, 3]]_p\) quantum Hamming code[1164] A family of CSS codes extending quantum Hamming codes to prime qudits of dimension \(p\) by expressing the qubit code stabilizers in local-dimension-invariant (LDI) form [1164].
- \([[2^r-1,1,3]]\) simplex code[1158,1165,1166] a.k.a. \([[2^r-1,1,3]]\) quantum RM code.Member of color-code code family constructed with a punctured first-order RM\((1,m=r)\) \([2^r-1,r+1,2^{r-1}-1]\) code and its even subcode for \(r \geq 3\). Each code transversally implements a diagonal gate at the \((r-1)\)st level of the Clifford hierarchy [1166,1167]. Each code is a color code defined on a simplex in \(r-1\) dimensions [22,1168], where qubits are placed on the vertices, edges, and faces as well as on the simplex itself.
- \([[2^{2r-1}-1,1,2^r-1]]\) quantum punctured Reed-Muller code[1169; Ch. 7] Member of CSS code family constructed with a punctured self-dual RM \([2^r-1,2^{r-1},\sqrt{2}^{r-1}-1]\) code and its even subcode for \(r \geq 2\).
- \([[2m,2m-2,2]]\) error-detecting code[82,1170] a.k.a. Iceberg code.Self-complementary CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [1171; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [282].
- \([[3, 1, 3;2]]\) EA code[314] Distance-three EA stabilizer code encoding one logical qubit and using two ebits.
- \([[30,8,3]]\) Bring code[1172] a.k.a. Small stellated dodecahedron code.A \([[30,8,3]]\) hyperbolic surface code on a quotient of the \(\{5,5\}\) hyperbolic tiling called Bring's curve. Its qubits and stabilizer generators lie on the vertices of the small stellated dodecahedron. Admits a set of weight-five stabilizer generators.
- \([[3k + 8, k, 2]]\) triorthogonal code[1025; Appx. B] Member of the \([[3k + 8, k, 2]]\) family (for even \(k\)) of triorthogonal and quantum divisible codes that admit a transversal \(T\) gate and are relevant for magic-state distillation.
- \([[4,1,1,2]]\) Four-qubit subsystem code[91,92] Error-detecting four-qubit subsystem stabilizer code encoding one logical qubit and one gauge qubit.
- \([[4,2,2]]\) Four-qubit code[870,1173] a.k.a. \(C_4\) code.Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error.
- \([[4,2,2]]_{G}\) four group-qudit code[35][345; Sec. VIII] \([[4,2,2]]_{G}\) group quantum code that is an extension of the four-qubit code to group-valued qudits.
- \([[49,1,5]]\) triorthogonal code[1025; Appx. B] Triorthogonal and quantum divisible code which is the smallest distance-five stabilizer code to admit a transversal \(T\) gate [291,1025,1174]. Its \(X\)-type stabilizers form a triply even linear binary code in the symplectic representation.
- \([[5,1,2]]\) rotated surface code[551; Ex. 5] Rotated surface code on one rung of a ladder, with one qubit on the rung, and four qubits surrounding it.
- \([[5,1,3]]_q\) Galois-qudit code[216] True stabilizer code that generalizes the five-qubit perfect code to Galois qudits of prime-power dimension \(q=p^m\). It has \(4(m-1)\) stabilizer generators expressed as \(X_{\gamma} Z_{\gamma} Z_{-\gamma} X_{-\gamma} I\) and its cyclic permutations, with \(\gamma\) iterating over basis elements of \(GF(q)\) over \(GF(p)\).
- \([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code[749] An analog stabilizer version of the five-qubit perfect code, encoding one mode into five and correcting arbitrary errors on any one mode.
- \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code[842,1175] Modular-qudit stabilizer code that generalizes the five-qubit perfect code using properties of the multiplicative group \(\mathbb{Z}_q\) [1175]; see also [842; Thm. 13]. It has four stabilizer generators consisting of \(X Z Z^\dagger X^\dagger I\) and its cyclic permutations.
- \([[6,1,3]]\) Six-qubit stabilizer code[1176]
- \([[6,2,2]]\) \(C_6\) code[1177] Error-detecting self-dual CSS code used in concatenation schemes for fault-tolerant quantum computation. A set of stabilizer generators is \(IIXXXX\) and \(XXIIXX\), together with the same two \(Z\)-type generators.
- \([[6,2,3]]_{q}\) code[416,1178]
- \([[6,4,2]]\) error-detecting code[82,1170,1180,1181] Error-detecting six-qubit code with rate \(2/3\) whose codewords are cat/GHZ states. A set of stabilizer generators is \(XXXXXX\) and \(ZZZZZZ\). It is the unique code for its parameters, up to local equivalence [282; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [1182].
- \([[6k+2,3k,2]]\) Campbell-Howard code[830] Family of \([[6k+2,3k,2]]\) qubit stabilizer codes with quasi-transversal \(CZZ^{\otimes k}\) gates that are relevant to magic-state distillation.
- \([[7, 1:1, 3]]\) hybrid stabilizer code[537] A distance-three seven-qubit hybrid stabilizer code storing one qubit and one classical bit. Admits a stabilizer generator set with a weight-two generator, which delineates the underlying classical code [317; Eq. (3)].
- \([[7,1,3]]\) Steane code[1183] A \([[7,1,3]]\) self-dual CSS code that is the smallest qubit CSS code to correct a single-qubit error [1176]. The code is constructed using the classical binary \([7,4,3]\) Hamming code for protecting against both \(X\) and \(Z\) errors.
- \([[7,1,3]]\) twist-defect surface code[1024] A \([[7,1,3]]\) code (different from the Steane code) that is a small example of a twist-defect surface code.
- \([[7,3,3]]_{q}\) code[416,1178]
- \([[8, 2:1, 3]]\) hybrid stabilizer code[537] A code obtained from the \([[8,3,3]]\) Gottesman code by using one of its logical qubits as a classical bit. One can also use two logical qubits as classical bits, obtaining an \([[8,1:2,3]]\) hybrid stabilizer code.
- \([[8, 3, 3]]\) Eight-qubit Gottesman code[82,873,1163] Eight-qubit non-degenerate code that can be obtained from a modified CSS construction using the \([8,4,4]\) extended Hamming code and a \([8,7,2]\) even-weight code [82]. The modification introduces signs between the codewords.
- \([[8,2,2]]\) hyperbolic color code[544] An \([[8,2,2]]\) hyperbolic color code defined on the projective plane.
- \([[8,3,2]]\) CSS code[1160,1161] a.k.a. Smallest interesting color code.Smallest 3D color code whose physical qubits lie on vertices of a cube and which admits a (weakly) transversal CCZ gate.
- \([[9,1,3]]\) Shor code[91] Nine-qubit CSS code that is the first quantum error-correcting code.
- \([[9,1,3]]_{\mathbb{R}}\) Lloyd-Slotine code[748,749] An analog stabilizer version of Shor's nine-qubit code, encoding one mode into nine and correcting arbitrary errors on any one mode.
- \([[9,1,3]]_{\mathbb{Z}_q}\) modular-qudit code[1184] Modular-qudit CSS code that generalizes the \([[9,1,3]]\) Shor code using properties of the multiplicative group \(\mathbb{Z}_q\).
- \([[9,1,5]]_3\) quantum Glynn code[1185] Nine-qutrit pure Hermitian code that is the smallest qutrit stabilizer code to correct two-qutrit errors.
- \([[9m-k,k,2]]_3\) triorthogonal code[797] Member of the \([[9m-k,k,2]]_3\) family of triorthogonal qutrit codes (for \(k\leq 3m-2\)) that admit a transversal diagonal gate in the third level of the qudit Clifford hierarchy and that are relevant for magic-state distillation.
- \([[k+4,k,2]]\) H code[1186] Family of \([[k+4,k,2]]\) self-dual CSS codes (for even \(k\)) with transversal Hadamard gates that are relevant to magic state distillation. The four stablizer generators are \(X_1X_2X_3X_4\), \(Z_1Z_2Z_3Z_4\), \(X_1X_2X_5X_6...X_{k+4}\), and \(Z_1Z_2Z_5Z_6...Z_{k+4}\).'
- \(\Lambda_{16}\) Barnes-Wall lattice code[98] BW lattice in dimension \(16\).
- \(\Lambda_{16}\) lattice-shell code Spherical code whose codewords are points on the \(\Lambda_{16}\) Barnes-Wall lattice normalized to lie on the unit sphere.
- \(\Lambda_{24}\) Leech lattice code[228] Even unimodular lattice in 24 dimensions that exhibits optimal packing. Its automorphism group is the Conway group \(.0\) a.k.a. Co\(_0\).
- \(\Lambda_{24}\) Leech lattice-shell code[228] Spherical code whose codewords are points on the \(\Lambda_{24}\) Leech lattice normalized to lie on the unit sphere. The minimal shell of the lattice yields the \((24,196560,1)\) code, and recursively taking their kissing configurations yields the \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes [950]; all codes are optimal and unique for their parameters [1096,1097].
- \(\chi^{(2)}\) code[1187] A \(3n\)-mode bosonic Fock-state code that requires only linear optics and the \(\chi^{(2)}\) optical nonlinear interaction for encoding, decoding, and logical gates. Codewords lie in Fock-state subspaces that are invariant under Hermitian combinations of the \(\chi^{(2)}\) nonlinearities \(abc^\dagger\) and \(i abc^\dagger\), where \(a\), \(b\), and \(c\) are lowering operators acting on one of the \(n\) triples of modes on which the codes are defined. Codewords are also \(+1\) eigenstates of stabilizer-like symmetry operators, and photon parities are error syndromes.
- \(\mathbb{Z}^n\) hypercubic lattice code Lattice-based code consisting of all integer vectors in \(n\) dimensions. Its generator matrix is the \(n\)-dimensional identity matrix. Its automorphism group consists of all coordinate permutations and sign changes.
- \(\mathbb{Z}_3\times\mathbb{Z}_9\)-fusion subsystem code[165] Modular-qudit 2D subsystem stabilizer code whose low-energy excitations realize a non-modular anyon theory with \(\mathbb{Z}_3\times\mathbb{Z}_9\) fusion rules. Encodes two qutrits when put on a torus.
- \(\mathbb{Z}_q^{(1)}\) subsystem code[165,1188] Modular-qudit subsystem code, based on the Kitaev honeycomb model [521] and its generalization [1188], that is characterized by the \(\mathbb{Z}_q^{(1)}\) anyon theory [1189], which is modular for odd prime \(q\) and non-modular otherwise. Encodes a single \(q\)-dimensional qudit when put on a torus for odd \(q\), and a \(q/2\)-dimensional qudit for even \(q\). This code can be constructed using geometrically local gauge generators, but does not admit geometrically local stabilizer generators. For \(q=2\), the code reduces to the subsystem code underlying the Kitaev honeycomb model code as well as the honeycomb Floquet code.
- \(k\)-orthogonal code[544,675,1190] Qubit stabilizer code whose \(X\)-type logicals and generators form a \(k\)-orthogonal matrix (defined below) in the symplectic representation. In other words, the overlap between any \(k\) \(X\)-type code-preserving Paulis (including the identity) is even. The original definition is for qubit CSS codes [675], but it can be extended to more general qubit stabilizer codes [1190; Def. 1]. This entry is formulated for qubits, but an extension exists for modular qudits [675].
- \(q\)-ary Hamming code[441,1191] Member of an infinite family of perfect linear \(q\)-ary codes with parameters \([(q^r-1)/(q-1),(q^r-1)/(q-1)-r, 3]_q\) for \(r \geq 2\).
- \(q\)-ary LDGM code \(q\)-ary linear code with a sparse generator matrix. Alternatively, a member of an infinite family of \([n,k,d]_q\) codes for which the number of nonzero entries in each row and column of the generator matrix are both bounded by a constant as \(n\to\infty\).
- \(q\)-ary LDPC code[1192] a.k.a. Non-binary LDPC (NBDPC) code.A \(q\)-ary linear code with a sparse parity-check matrix. Alternatively, a member of an infinite family of \([n,k,d]_q\) codes for which the number of nonzero entries in each row and column of the parity-check matrix are both bounded above by a constant as \(n\to\infty\).
- \(q\)-ary code Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\), i.e., \(q\)-ary strings. Error-correcting performance is quantified by some distance \(d\), which in turn is defined using a metric. The default distance is the Hamming distance \(d\), the weight (i.e., number of nonzero coordinates) of the lowest-weight nonzero codeword; such codes are usually denoted as \((n,K,d)_q\). The corresponding Hamming metric between two \(q\)-ary strings is the number of coordinates in which they differ. Unless stated otherwise, the distance for this class is the Hamming distance.
- \(q\)-ary code over \(\mathbb{Z}_q\) A code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_q\) of integers modulo \(q\).
- \(q\)-ary duadic code[119,1193–1195] Member of a pair of cyclic linear binary codes that satisfy certain relations, depending on whether the pair is even-like or odd-like duadic. Duadic codes exist only when \(q\) is a square modulo \(n\) [119].
- \(q\)-ary linear LCC A linear code for which one can recover any coordinate of a codeword from at most \(r\) coordinates of the error word (assuming the error word is within some tolerated corruption rate \(\delta\)).
- \(q\)-ary linear LTC A \(q\)-ary linear code \(C\) of length \(n\) that is a \((u,R)\)-LTC with query complexity \(u\) and soundness \(R>0\). More technically, the code is a \((u,R)\)-LTC if the rows of its parity-check matrix \(H\in GF(q)^{r\times n}\) have weight at most \(u\) and if \begin{align} \frac{1}{r}|H x| \geq \frac{R}{n} D(x,C) \tag*{(14)}\end{align} holds for any \(q\)-ary string \(x\), where \(D(x,C)\) is the \(q\)-ary Hamming distance between \(x\) and the closest codeword to \(x\) [488; Def. 11]. A code satisfying the above constraint without the weight-\(u\) restriction is called an \(R\)-testable code [1196].
- \(q\)-ary linear code over \(\mathbb{Z}_q\) A linear code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_q\) of integers modulo \(q\).
- \(q\)-ary parity-check code a.k.a. Sum-zero code, Zero-sum code.An \([n,n-1,2]_q\) linear \(q\)-ary code whose codewords consist of the message string appended with a parity-check or zero-sum check digit such that the sum over all coordinates of each codeword is zero.
- \(q\)-ary protograph LDPC code[1197–1200] A \(q\)-ary LDPC code whose parity-check matrix is constructed using the lifting procedure applied to the incidence matrix of a sparse graph called, in this context, a protograph. An ability to assign non-binary edge weight called edge scaling can also be used in code construction.
- \(q\)-ary quadratic-residue (QR) code
- \(q\)-ary repetition code An \([n,1,n]_q\) code encoding consisting of codewords \((j,j,\cdots,j)\) for \(j \in GF(q)\). The length \(n\) needs to be an odd number, since the receiver will pick the majority to recover the information.
- \(q\)-ary sharp configuration[226,922,1073] A \(q\)-ary code that admits \(m\) different distances between distinct codewords and forms a design of strength \(2m-1\) or greater.
- \(q\)-ary simplex code[337,1146] a.k.a. \(q\)-ary maximum-length feedback-shift-register code.An \([n,m,q^{m-1}]_q\) projective code with \(n=\frac{q^m-1}{q-1}\), denoted as \(S(q,m)\). The columns of the generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,q)\), with each column being a chosen representative of the corresponding element.
- \(t\)-design a.k.a. Cubature, Averaging set.A code whose codewords are uniformly distributed in a way that is useful for determining averages of polynomials over the code's underlying space \(X\). In that way, the codewords form an approximation of the space. A code is a design on \(X\) of strength \(t\), i.e., a \(t\)-design on \(X\), if the average of any polynomial of degree up to \(t\) over its codewords is equal to the uniform average over all of \(X\).
- \(t\)-erasure LRC a.k.a. Multiple-erasure LRC.A code which admits local recoverability against more than one coordinate erasure.
- Æ code[1203] Code defined in a single angular-momentum subspace that is embedded in a larger direct-sum space of different angular momenta, which can arise from combinations of spin, electronic, or rotational, or nuclear angular momenta of an atom or molecule. A code is obtained by solving an over-constrained system of equations, and many solutions can be mapped into existing codes defined on other state spaces.

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