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.

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.

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].

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.

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).

Lattice stabilizer code in two spatial dimensions.

A subsystem version of the 2D color code.

Two-mode qutrit code constructed out of superpositions of coherent states whose amplitudes make up the binary tetrahedral group \(2T\), a.k.a. the 24-cell.

Generalization of the Bacon-Shor code to three dimensions that was conjectured to be a self-correcting memory. It is defined on a cubic lattice and admits sheet-like stabilizer generators.

3D subsystem stabilizer code whose Hamiltonian is a 3D generalization of the Kitaev honeycomb model. One of the phases realized by the 3D Kitaev honeycomb Hamiltonian is that of the 3D fermionic surface code [15].

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.

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).

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 [20]. The model can be defined on a cubic lattice in several ways [21; Eq. (D45-46)]. Realizations on other lattices also exist [22], and the phase of this code also exists in the 3D Kitaev honeycomb model [15].

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 [21].

A subsystem version of the 3D color code.

Subsystem generalization of the surface code on a 3D cubic lattice with gauge-group generators of weight at most three.

A generalization of the Kitaev surface code defined on a 3D lattice.

Spherical \((4,120,(3-\sqrt{5})/2)\) code whose codewords are the vertices of the 600-cell. See [27; 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 \(2I\) [28].

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) [30; Sec. III.E]. A particular family with \(G=\mathbb{Z}_{\ell}\) yields codes with constant rate and nearly constant distance.

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 [34; Sec. IV.A]. Abelian TQDs realize all modular gapped Abelian topological orders [34]. Many Abelian TQD code Hamiltonians were originally formulated as commuting-projector models [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.

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 [37]. Unless otherwise noted, the phases discussed are bosonic.

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.

A \(q\)-ary code whose codewords are closed under addition, i.e., for any codewords \(x,y\), \(x+y\) is also a codeword.

The simplest OSTBC, with \(n=2\) transmitting antennas, \(m=1\) receiving antennas, and \(t=2\) uses.

LDPC code whose parity check matrix is constructed explicitly (i.e., non-randomly) from a particular graph [40,41] or an algebraic structure such as a combinatorial design [42–44], balanced incomplete block design [45], a partial geometry [46], a generalized polygon [47,48], or a Latin square [49–51]. The extra structure and/or symmetry [52] of these codes can often be used to gain a better understanding of their properties.

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.

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 [60; Ch. 12]. Its parity-check matrix is an alternant matrix.

Block quantum code on either qubits or bosonic modes that is designed to detect and correct qubit or bosonic AD errors, respectively.

Self-complementary CWS code that is designed to detect and correct AD errors.

A block code over the real or complex numbers, called a real RS or complex RS (CRS) code respectively, whose encoding generalizes the RS code encoding over finite fields.

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 constructions have to be considered.

Encodes states (codewords) into coordinates in the \(n\)-dimensional (real or complex) coordinate space (\(\mathbb{R}^n\) or \(\mathbb{C}^n\)). Codes for storing discrete sets of symbols include sphere packings, tilings, and any other codes that use real or complex numbers for encoding. The number of codewords may be infinite because the coordinate space is infinite, so various restricted versions have to be constructed in practice.

An \([[n,1]]_{\mathbb{R}}\) analog stabilizer version of the quantum repetition code, encoding the position states of one mode into an odd number of \(n\) modes.

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. Any analog stabilizer state can be thought of as a pure Gaussian state that has been infinitely squeezed on all modes [71].

An analog CSS version of the Kitaev surface code realizing a phase of 2D \(\mathbb{R}\) gauge theory.

Code whose codewords are obtained from a simulated annealing or energy-repulsion numerical optimization procedure.

Code for which the distance between any two codewords is less than or equal to some value \(\delta\) called the maximum distance. Anticodes can be used to construct codes that saturate the Griesmer bound; see Refs. [60,78,79] for more details.

Sphere packing constructed via the antipode construction.

Code encoding quantum and/or classical information that approximately corrects against noise affecting operators forming an algebra.

Encodes quantum information so that it is possible to approximately recover that information from noise up to an error bound in recovery.

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.

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.

QC-LDPC code constructed deterministically from a disk array code known as a B-code. Its parity-check matrix admits a compact representation [89] and is related to RS codes.

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.

A concatenation of the JW transformation code with a qubit stabilizer code.

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.

The first array code, constructed over \(GF(q)\). See [97] for more details.

A BB code defined by weight-five measurements designed for long chains of trapped ions. Examples include \([[30,4,5]]\) and \([[48,4,7]]\) codes,

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 [99].

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.

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 \((\frac{1}{2}-\frac{1}{n^{\gamma}},\frac{1}{2}+\frac{1}{n^{\gamma}})\).

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.

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).

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.

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.

Binary code designed for minimizing the total amount of storage and the worst-case maximal load on any devices in a distributed system.

Locally testable \([n,k,d]\) code with \(n = k^{1+\epsilon}\) and query complexity of order \(O(1/\epsilon)\) for any \(\epsilon > 0\).

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 \(c\).

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 \(c\).

A linear \(p\)-ary code (for prime \(p\)) 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 [115; Ch. 10.6].

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.

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 \(r\geq 2\).

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.

An \((n,K,4d/n)\) spherical code obtained from a binary \((n,K,d)\) code via the antipodal mapping.

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 \(q\) [121; Sec. 6.6].

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.

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 \( P = \text{diag} ( 1, \omega^2) \).

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 \(8\) [124].

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.

A binary linear code \(C\) of length \(n\) that is a \((u,R)\)-LTC with query complexity \(u\) and soundness \(R>0\).

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 \(n\).

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.

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 [129].

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).

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.

Member of a particular class of irregular QC-LDPC codes with efficient encoders.

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 \(\Sigma\).

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.

Three-dimensional lattice consisting of all points \((x,y,z)\) whose integer components are either all even or all odd.

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 stabilizer code admitting a set of stabilizer generators that are either position or momentum displacements.

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 [134].

Bosonic code designed for transmission of classical information through non-classical channels.

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).

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 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. Stabilizer groups are any locally compact Abelian subgroups of \(\mathbb{R}^n\), can themselves contain discrete or continuous subgroups, and can admit logical qudit and/or oscillator logical subspaces.

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.

Code whose codewords are points lie on or inside a real or complex sphere whose radius squared is called the energy.

Qubit stabilizer code whose encoding circuit corresponds to a branching MERA tensor network [142].

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.

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 \(\geq 6\) [145].

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 [146], and the Pauli-string weight can be further optimized to yield the segmented Bravyi-Kitaev (SBK) transformation code [147].

An \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth of order \(O(\log^3 n)\).

Generalization of the Bacon-Shor code to three dimensions, defined on a cubic lattice and admitting string-like stabilizer generators.

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 [151].

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.

A Hermitian qubit QLDPC code whose stabilizer generator matrix is constructed using two nested subgroups of \(GF(4)^n\).

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 \(PG_{5}(q)\) [121; Exam. 9.4.5]. There exist multiple distinct spherical codes using this construction for \(q>3\) [154].

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).

A generalization of the Goppa codes to codes defined from curves of nonzero genus. Each code is a subcode of a certain residue AG code and is constructed using the Cartier operator.

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 \(\alpha\).

A concatenated qubit-into-\(n\)-mode code whose outer code is a quantum repetition code and whose inner code is the cat code in its cat basis.

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.

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 [21; Eq. (D38)].

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.

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.

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 [20,165]; see Ref. [166] for a different lattice-model formulation of the FcBl boundary code.

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 \(r = \deg G \leq n-1\).

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 [168,169]. The corresponding phase can also be realized via a non-stabilizer Hamiltonian [170].

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.

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.

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 \(\Omega(n/\log^5 n)\) [172; 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.

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 code defined on a two-dimensional lattice and derived from a geometrically local stabilizer code, such as the surface code or color code.

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 [179] 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 [180].

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.

Non-uniform 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.

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 \(SU(2)\).

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 [186] 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.

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.

A code based on a cluster state and often used in measurement-based quantum computation (MBQC) [189,190] (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.

Encodes codewords into permutations of \(n\) objects.

A code with \((r,\delta)\) locality is a code that encodes each codeword coordinate into an \([r+\delta-1,r,\delta]\) MDS code [193; 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 \(\delta\).

A code defined using a cluster state and a set of \(Z\)-type Pauli strings defined by a binary classical code.

Coherent-state c-q code encoding into coherent states that are frequency-shifted with certain initial relative phase.

A qubit stabilizer code for which two binary linear codes are used to directly construct encoding and decoding circuits against \(X\)- and \(Z\)-type errors, respectively, via ZX calculus [201,202]. CPC codes can be obtained from numerical search [199].

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.

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 [203,204].

A concatenated qubit-into-\(n\)-mode code (for odd \(n\)) whose outer code is a quantum repetition code and whose inner code is the two-component cat code in its coherent-state basis.

Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs.

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.

A constant-weight binary code that is mapped into a combinatorial \(t\)-design.

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.

An integer-homology bosonic CSS code realizing 2D \(U(1)\) gauge theory on bosonic modes. The code can be obtained from the analog surface code by condensing certain anyons [208]. This results in a pinning of each mode to the space of periodic functions, which make up a physical rotor, and can be thought of as compactification of the 2D \(\mathbb{R}\) gauge theory phase realized by the analog surface code.

Subspace or subsystem CSS code defined by gauge-fixing the Bacon-Shor code, i.e., the code whose gauge group consists of terms in the compass model Hamiltonian [210–212] on a square lattice. Families of random codes perform well against biased noise and spatially dependent (i.e., asymmetric) noise.

Evaluation code of polynomials evaluated on points lying on a complete intersection.

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 \(C\) [215].

A spherical code obtained from particular complex Hadamard matrices [217].

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.

A member of the family of \([[7^m,1,3^m]]\) CSS codes, each of which is a recursive level-\(m\) concatenation of the Steane code. This family is one of the first to admit a concatenated threshold [219–223].

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.

A c-q code constructed out of two classical or quantum codes for the purposes of transmission of classical information over quantum channels.

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 in its cat-state basis.

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.

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 [227].

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.

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.

Approximate code whose codewords lie in the low-energy subspace of a conformal field theory, e.g., the quantum Ising model at its critical point [229,231]. Its encoding is argued to perform source coding (i.e., compression) as well as channel coding (i.e., error correction) [229].

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.

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.

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 (and, similarly, for fermion) codes, such as Fock-state codes, codewords are often in an eigenspace with eigenvalue \(N>0\) of the total excitation or energy Hamiltonian, \(H=\sum_i \hat{n}_i\).

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 \(J(n,w)\) [235–238].

A nonlinear single-asymmetric-error code that generalize VT codes and that is constructed from an Abelian group.

Sphere packing constructed from a binary \((n,K)\) code using Construction A [241].

A lattice that is constructed from a linear code over \(\mathbb{Z}_4\) using Construction \(A_4\).

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.

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 [243] and error-corrected parameter estimation.

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.

Even integral lattice in dimension \(12\) that exhibits optimal packing. It's automorphism group was discovered by Mitchell [245]. For more details, see [246][241; Sec. 4.9].

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 [97].

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 \(D_4\).

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.

3D color code defined on a four-colorable bitruncated cubic honeycomb uniform tiling.

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.

An LDPC code whose parity-check matrix forms the incidence matrix of a graph, i.e., has weight-two columns.

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.

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.

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.

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.

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 \(\mathbb{Z}_n\).

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.

Code that was designed (or that can be applied) to encode information into the four-base-pair alphabet of a DNA molecule.

Evaluation code of polynomials evaluated on points lying on a Deligne-Lusztig curve.

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.

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.

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 [271; Table I]. It has been extended for models with several modes per site [273].

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 \(n\).

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 \(s = \text{lcm}(1,2,\cdots,r)\).

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.

Cyclic LDPC code constructed deterministically from a difference set. Certain DCS codes satisfy more redundant constraints than Gallager codes and thus can outperform them [278].

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 [36]. An alternative qubit-based formulation realizes the gauged \(G=\mathbb{Z}_3^2\) twisted quantum double phase [279], which is the same topological order as the \(G=D_4\) quantum double [280,281].

A code whose codewords realize \(D\)-dimensional lattice Dijkgraaf-Witten gauge theory [282,283] 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 [284] as well as explicitly in 3D [285,286].

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 [288].

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.

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 (\(t=2\)).

Spherical \((4,48,2-\sqrt{2})\) code [121; Exam. 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 \(2O\) [28; Sec. 8.6].

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 [291], later generalized [293; Thm. 4.2], can yield QLDPC codes [291; Thm. 1].

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.

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.

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) [241,295]. 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.

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 [297; Sec. 2.4.8]. Another generator matrix can be found in [298; Exam. 9.10.8].

Spherical \((3,20,2-2\sqrt{5}/3)\) code whose codewords are the vertices of the dodecahedron (alternatively, the centers of the faces of a icosahedron, the dodecahedron's dual polytope).

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 \(e^2 m^2\) [171]. Originally formulated as the ground-state space of a Hamiltonian with non-commuting terms [299], which can be extended to other spatial dimensions [300], and later as a commuting-projector code [35,301].

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 [305], that admit a Clifford + \(T\) transversal gate set using gauge fixing.

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*{(2)}\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.

For any dimensional lattice \(L\), the dual lattice is the set of vectors whose inner products with the elements of \(L\) are integers.

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.

For any linear code \(C\) over a ring \(R\), the dual code is the set of strings that are orthogonal to the codewords of \(C\) under a particular inner product.

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.

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.

Dynamically generated stabilizer-based code whose (not necessarily periodic) sequence of few-body measurements implements state initialization, logical gates and error detection.

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.

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.

Galois-qudit code designed to utilize pre-shared entanglement between sender and receiver.

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 Galois-qudit code whose parameters make the EAQECC Singleton bound (a.k.a. qubit-ebit Singleton bound) [316; Thm. 6] become an equality.

One of several EA QLDPC code families constructed from QC-LDPC codes.

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 \(n\to\infty\)

Constructed using a variation of the analog stabilizer formalism designed to utilize pre-shared entanglement between sender and receiver.

Bosonic code designed to utilize pre-shared entanglement between sender and receiver.

One of several EA QLDPC code families constructed from combinatorial designs.

An EA Galois-qudit stabilizer code constructed from an LCD code. This family include the first asymptotically good EA Galois-qudit codes.

A quantum convolutional code designed to utilize pre-shared entanglement between sender and receiver, which can reduce memory requirements [323].'

A quantum turbo code which uses pre-shared entanglement. This allows its encoder to be both recursive and non-catastrophic.

Qubit code designed to utilize pre-shared entanglement between sender and receiver.

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.

A binary array code that can correct any two disk failures (i.e., two erasures). See [97] for more details.

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.

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.

Evaluation AG code of rational functions evaluated on points lying on an elliptic curve, i.e., a curve of genus one.

QECC whose encoding and decoding utilizes pre-shared entanglement between sender and receiver.

Code that encodes quantum and classical information and requires pre-shared entanglement for transmission.

Subsystem QECC whose encoding and decoding utilizes pre-shared entanglement between sender and receiver.

A code family that encompasses ordinary (i.e., subspace) codes, subsystem codes, classical-quantum codes, hybrid codes, and their entanglement-assisted counterparts using a unified operator-algebraic framework.

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.

Code designed for transmission of classical information through classical channels in a robust way.

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. The \(\pm 1\) elements can be used to distinguish between a pair of chosen classes, while a zero entry correspond to a classifier ignoring that particular class.

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.

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.

Family of \(G\)-lifted product codes constructed using two random classical Tanner codes defined on expander graphs [345]. 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.

LDPC code whose parity-check matrix is derived from the adjacency matrix of bipartite expander graph [345] such as a Ramanujan graph or a Cayley graph of a projective special linear group over a finite field [347,348]. Expander codes admit efficient encoding and decoding algorithms and yield an explicit (i.e., non-random) asymptotically good LDPC code family.

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.

A generalization of the IRA code in which the outer LDGM code is replaced by a random sparse matrix containing no weight-two columns.

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 [144]. Majorana operators may either be considered individually or paired in various ways into creation and annihilation operators to yield fermionic modes. They form a Clifford algebra and can be interpreted as Ising anyons in certain contexts.

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.

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.

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.

A fractal type-I fracton CSS code defined on a cubic lattice [21; Eq. (D23)].

Quantum error correcting code associated with the Levin-Wen string-net model with the Fibonacci input category, admitting two types of encodings.

An error-correcting code defined over a finite alphabet.

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.

CSS code constructed from linear binary codes whose parity-check or generator matrices are incidence matrices of points, hyperplanes, or other structures in finite geometries. These codes can be interpreted as quantum versions of FG-LDPC codes, but some of them [357,358] are not strictly QLDPC.

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 [360].

Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error.

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.

Evaluation code of polynomials evaluated on points lying on a flag variety.

3D Floquet code whose physical qubits are placed on vertices of a particular trivalent honeycomb. Its weight-two check operators are those of the 3D Kitaev honeycomb model [15]. Its ISGs are equivalent to that of the 3D fermionic surface code via a constant-depth circuit with ancillas.

3D Floquet code whose physical qubit pairs are placed on vertices of a truncated cubic honeycomb. The code is constructed by coupling three 2D surface codes on three interweaving 4.8.8 (square-octagon) tilings oriented along the \(x\), \(y\), and \(z\) axes, respectively, via a coupled layer construction [366]. Its ISG is that of the 3D surface code.

2D 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.

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.

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*{(3)}\end{align}

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.

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 PI code that is the smallest qubit code to correct one deletion error.

\([[4,2,2]]_{\mathbb Z}\) 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.

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 [375]. The underlying classical codes form classical self-correcting memories [378–380].

3D 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.

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.

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 [384].

Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [386]. 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 [387]. 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.

A \(q\)-ary frequency-shift keying (\(q\)-FSK) encodes one \(q\)-ary digit of information into signals with \(q\) different frequencies.

A cyclic prime-qudit stabilizer code whose length \(n\) divides \(p^t + 1\) for some positive integer \(t\).

Hamiltonian-based code whose Hamiltonian is frustration free, i.e., whose ground states minimize the energy of each term.

Code whose codewords are resource states used in an FBQC scheme. Related to a cluster state via Hadamard transformations.

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 [390–393].

A concatenated code whose outer code is a GKP code and whose inner code is a toric surface code [394], rotated surface code [392,395–398], or XZZX surface code [399].

PI code whose codewords can be expressed as superpositions of Dicke states with coefficients are square-roots of the binomial distribution.

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.

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.

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 [416], e.g., as in Ref. [413].

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.

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.

True \(q\)-Galois-qudit stabilizer code constructed from GRS codes via either the Hermitian construction [425–427] or the Galois-qudit CSS construction [424,428].

A member of a family of Galois-qudit CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear \(q\)-ary codes.

An \([[n,k,n-k+1]]_q\) (with \(q>n\)) Galois-qudit CSS code constructed using two RS codes over \(GF(q)\).

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.

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 \(p\).

Extension of the color code to 2D lattices of Galois qudits.

Galois-qudit CSS code constructed from a hypergraph product of expander codes.

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 [438; Sec. 4.2].

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.

Extension of the surface code to 2D lattices of Galois qudits.

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 [444; 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 [443,444].

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 \(G=\mathbb{Z}_2\times\mathbb{Z}_2\)).

A generalization of the EVENODD code defined for parameters \(m\) and \(r\); see [97].

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.

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 \(GF(q)\).

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.

Qubit CSS code constructed by concatenating two classical codes in a way the generalizes the Shor and quantum parity codes.

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 \(GF(q^m)\).

A quasi-cyclic Galois-qudit CSS code constructed using a generalized version of the bicycle ansatz [116] 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 [30; Sec. III.E].

A code that combines multiple outer codes of the same length and (possibly) different dimensions with a single inner code; see Refs. [457][60; Ch. 18].

Member of a family of subsystem codes that are generalizations [459,460] of a code defined on a three-valent hypergraph associated with the five-squares lattice [458].

CSS code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes.

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.

Qubit CSS code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes.

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.

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 \(t\) [462; Def. V.1]. Such codes admit gates at the \(\nu\)th level of the Clifford hierarchy. Such codes can also be level-lifted [462; Theorem V.6], \(\nu\to\nu+1\), which recursively yields towers of generalized divisible codes from a particular ground code.

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.

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 [60; Ch. 15].

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 \(r\) [465].

Variant of the Guth-Lubotzky hyperbolic surface code that uses regular tessellations for 4-dimensional hyperbolic space.

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 \(k^{1+o(1)}\).

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.

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 \( R_a(x) = \sum_{i=1}^n \frac{a_i}{z - \alpha_i} \).

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 \(A\).

A stabilizer code on tensor products of \(G\)-valued qudits for Abelian \(G\) whose encoding isometry is defined using a graph [473; 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 [473]; see [474; 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 [475].

Binary linear code whose generator matrix is that of the row space of 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.

Evaluation code of polynomials evaluated on points lying on a Grassmannian \({\mathbb{G}}(\ell,m)\) [481].

The first Gray code [482], 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 via what is known as the Gray map.

A type of \(q\)-ary code whose parameters satisfy the Griesmer bound with equality.

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.

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 \(GF(q)^n\) [490][489; Lemma 2.3]. A group-algebra code for an Abelian group is called an Abelian group-algebra 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.

An \([[m r,1,\min(m,r)]]_G\) generalization of the QPC.

A code based on a group-based cluster state for a finite group \(G\) [491]. 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.

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 or \(G\)-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.

An \([[n,1]]_G\) generalization of the quantum repetition 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.

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.

Extension of the Kitaev surface code from Abelian groups to groupoids, i.e., multi-fusion categories in which every morphism is an isomorphism [494]. 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.

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.

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 codewords are generated in a process involving averaging over unitary operations distributed according to the Haar measure. Haar-random codes are used to prove statements about the capacity of a quantum channel to transmit quantum information [500], but encoding and decoding in such \(n\)-qubit codes quickly becomes impractical as \(n\to\infty\).

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.

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\(^*(1,m)\)).

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.

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.

A member of a family of extremal Type II self-dual codes over \(\mathbb{Z}_4\) that yield all Niemeier lattices via Construction \(A_4\).

2D dynamical code whose sequence of check-operator measurements is periodic. The first instance of a dynamical code.

An \([[n,1]]_{\mathbb{R}}\) analog CSS code defined using homological structures associated with an \(n-1\) simplex. Relevant to the study of spacetime replication of quantum information [505].

Subsystem stabilizer code on the heavy-hexagonal point set 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 point set 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 [507].

Homological code constructed out of cubes in high dimensions. The hemicubic code family has asymptotically diminishing soundness that scales as order \(\Omega(1/\log n)\), locality of stabilizer generators scaling as order \(O(\log n)\), and distance of order \(\Theta(\sqrt{n})\).

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.

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.

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 \(GF(q^2)\).

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 \(q^3 + 1\).

An \([[n,k,d]]\) stabilizer code constructed from a Hermitian self-orthogonal linear quaternary code using the \(GF(4)\) representation.

Evaluation code of polynomials evaluated on points lying on a Hermitian hypersurface.

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 [121; 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 [154,517–519].

The \([6,3,4]_4\) self-dual MDS code that has connections to projective geometry, lattices [241], and conformal field theory [521]. Puncturing the code yields the perfect \([5,3,3]_4\) quaternary Hamming code known as the shortened hexacode or shorter hexacode [522]. Both codes are sometimes refereed to as Golay codes over \(GF(4)\).

Single-mode GKP qudit-into-oscillator code based on the triangular lattice. Offers the best error correction against displacement noise in a single mode due to the optimal packing of the underlying lattice.

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 [524,525].

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 [347,348]. 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.

A graph-based code whose generator matrix is constructed using the adjacency matrix of the Higman-Sims graph. This family includes a \([100,22,32]\) code whose dual is an optimal \([100,78,8]\) code [476; Table VI].

Member of a projective code family that contains of \(q\)-ary sharp configurations and that is constructed using projective caps.

A projective geometry code that is an example of an MDS code that is not an RS code; see [529; Exam. 7.6] for the description.

A \([50,21,12]\) cycle code whose parity-check matrix is the incidence matrix of the Hoffman-Singleton graph [530]. Its dual is a \([50,29,8]\) code [476; Table II].

A graph-based code whose generator matrix is that of the row space of the adjacency matrix of the Hoffman-Singleton graph [530]. This \([50,22,7]\) code is dual to a \([50,28,8]\) code [476; Table III].

Block quantum code whose features serve to model aspects of the AdS/CFT holographic duality and, more generally, quantum gravity.

Holographic tensor-network code constructed out of alternating isometries of the five-qubit and \([[4,1,1,2]]\) Bacon-Shor codes.

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.

CSS-type extension 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.

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.

CSS code formulated using the tensor product of two chain complexes of length one or greater (see Qubit CSS-to-homology correspondence).

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.

2D color code defined on a patch of the 6.6.6 (honeycomb) tiling.

2D Floquet code based on the Kitaev honeycomb model [541] 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 [368]. The code has also been generalized to arbitrary non-chiral, Abelian topological order [542].

A two-dimensional point set whose points are vertices of hexagons. It is not a lattice since its points do not form a group under addition. As a tiling, its dual (whose points lie at the centers of each triangle) is the triangular tiling. The ruby tiling is a fattened honeycomb tiling interpolating between the honeycomb tiling and triangular lattice.

Code based on a cluster state defined on qudits valued in a Hopf algebra. This code has also been extended to weak Hopf algebras [544].

Code whose codewords realize 2D gapped topological order defined on qudits valued in a Hopf algebra \(H\). The code Hamiltonian is an generalization [545,546] of the quantum double model from group algebras to Hopf algebras, as anticipated by Kitaev [36]. Boundaries of these models have been examined [547,548].

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 [21].

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 [21; Eqs. (D42-D43)].

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.

Homological code constructed on triangulations of Hurwitz surfaces.

A quantum code which encodes both quantum and classical information.

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.

A quantum convolutional code which protects both quantum and classical information.

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.

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 [560].

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.

Floquet code whose check-operators correspond to edges of a hyperbolic lattice of degree 3.

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 [565]. Certain double covers of hyperbolic tilings also yield admissible tilings [564]. Other admissible hyperbolic tilings can be obtained via a fattening procedure [8]; see also a construction based on the more general quantum pin codes [566].

An evaluation code over polynomials in two variables. Generator matrices are determined in Ref. [569] following initial formulations of the codes as generalized concatenations of RS codes [567,568]; see [341; Exam. 4.26].

Encodes states (codewords) into coordinates in the hyperbolic plane \(\mathbb{H}^2\).

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.

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.

A member of a family of CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear binary codes. Codes from hypergraph products in higher dimension are called higher-dimensional HGP codes [574].

Holographic tensor-network error-detecting code constructed out of a hyperinvariant tensor network [576], 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).

A projective code constructed using hyperovals in projective space.

Homological code based on products of hyperspheres. The hypersphere product code family has asymptotically diminishing soundness that scales as order \(O(1/\log (n)^2)\), locality of stabilizer generators scaling as order \(O(\log n/ \log\log n)\), and distance of order \(\Theta(\sqrt{n})\).

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).

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.

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 [583,584][585; Sec. 14.4]. Columns of a code's parity-check matrix can similarly correspond to an incidence matrix.

A bosonic stabilizer code whose physical modes have been restricted, via a single GKP stabilizer, from the space of functions on the real line to the space of periodic functions. This restriction effectively realizes a rotor on each physical mode, allowing one to construct homological rotor codes out of displacement stabilizer groups. The stabilizer group is continuous, but contains discrete components in the form of the single-mode GKP stabilizers. 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.

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*{(6)}\end{align}

An LDPC code whose parity-check matrix has a variable number of entries in each row or column.

A convolutional code whose parity-check matrix has a variable number of entries in each row or column.

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.

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 \(O(n)\).

One of several nonlinear binary \((12,144,4)\) codes based on the Steiner system \(S(5,6,12)\) [599,600][60; Sec. 2.7][601; 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.

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 \(w\).

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.

Member of the family of \([2m+2,m+1]\) double circulant codes such that \(m\) is prime of the form \(8k+3\) for some \(k\), and \(2m+2\) is a multiple of eight. See [60; Ch. 16] for their generator matrix.

Member of the family of \([2^{2r}-1, 3r, 2^{2r-1} - 2^{r-1} ]\) cyclic binary linear codes.

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\((2,m)\).

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 [121; 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 [612].

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.

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). It can be thought of as the Majorana stabilizer analogue of the quantum repetition code, and it encodes a logical fermion because its logical Majorana operator has odd weight [614].

Member of the family of \([[2n,(0,2),(2,n)]]_{\mathbb{Z}}\) homological rotor codes storing a logical qubit on a thin Möbius strip. The ideal code can be obtained from a Josephson-junction [617] system [540].

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 [541]. Ising anyons also exist in other phases, such as the fractional quantum Hall phase [619].

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.

Evaluation AG code over \(GF(8)\) of rational functions evaluated on points lying on the Klein quartic, which is defined by the equation \(x^3 y + y^3 z + z^3 x = 0\) ([341], Exam. 2.75).

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.

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 [624] showed that related concatenated codes achieve the GV bound.

Convolutional code defined by an infinite low-density parity-check matrix.

Code constructed using a 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.

1D 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.

Spherical code whose codewords are obtained from a recursive procedure that is similar to the procedure that creates laminated lattices.

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.

A geometrically local stabilizer code with sites 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-type 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.

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 subsystem stabilizer code Hamiltonians described by an Abelian anyon theory do not always realize the corresponding anyonic topological order in their ground-state subspace and may exhibit a rich phase diagram.

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.

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.

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.

LDPC code whose Tanner graph comes from a particular family of \(q\)-regular graphs [635] of known girth and relatively large stopping sets.

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 [639].

Binary code constructed on a left-right Cayley complex using a pair of base codes \(C_A,C_B\) and an expander graph [345] 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.

Binary codes constructed from combining two codes \(A'\) constructed out of Hadamard matrices.

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.

Member of one of several families of lifted-product codes that consist of sparsely interconnected stacks of surface codes.

Galois-qudit 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.

Spacetime code whose set of matrix codewords is closed under addition and subtraction.

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.

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.

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.

A code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_4\) of integers modulo four that is an additive group. More technically, linear codes over \(\mathbb{Z}_4\) are submodules of \(\mathbb{Z}_4^n\).

A code encoding \(K\) states (codewords) in \(n\) coordinates over the ring \(\mathbb{Z}_q\) of integers modulo \(q\) that forms an Abelian subgroup of \(\mathbb{Z}_q^n\) under addition. Since addition of \(m\) identical elements is equivalent to multiplying by \(m\), linear codes over \(\mathbb{Z}_q\) are automatically closed under scalar multiplication. More technically, linear codes over \(\mathbb{Z}_q\) are submodules of \(\mathbb{Z}_q^n\).

A linear code \(C\) admits a complementary dual if \(C\) and its dual code \(C^{\perp}\) do not share any codewords.

A linearized version of a skew RS code, i.e., an RS code constructed by evaluating skew polynomials [649,650].

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.

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 \(\delta\)).

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.

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.

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 \(R\).

A random \([[n,k]]\) stabilizer code whose encoder is a random Clifford circuit of depth of order \(O(\log n)\) on a 1D Euclidean geometry.

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 [662].

Code constructed using a 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.

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 [25,664].

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 [667] yields a \(c^3\)-LTC [665]. Using two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [666].

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.

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 \(n\to\infty\).

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.

Erasure codes based on fountain codes. They improve on random linear fountain codes by having a much more efficient encoding and decoding algorithm.

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.

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 \(g\) [673].

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 [674].

A Majorana stabilizer code which forms a fixed-parity subspace of the ground-state subspace of one or more Kitaev Majorana chain Hamiltonians. The \([[n,1,2]]_{f}\) Majorana box qubit forms the even-fermion-parity ground-state subspace of two parallel Kitaev Majorana chains in their fermionic topological phase. Its \([[2,1,2]]_{f}\) 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 [677].

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 analogue of the color code defined on a 2D tricolorable lattice and constructed out of Majorana box qubit codes or other small Majorana stabilizer codes placed on patches of the lattice.

An single error-correcting fermion-into-qubit encoding defined on a 2D qubit lattice whose stabilizers are associated with loops in the lattice.

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.

A Majorana stabilizer code with some of its logical qubits denoted as gauge qubits and not used for storage of logical information.

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.

Member of a class of LDPC codes deterministically constructed using an explicit class of regular graphs with no short cycles. Related explicit LDPC constructions [687] utilize Ramanujan graphs [347,348].

Member of a class of qubit stabilizer codes based on the Abelian phase of the Kitaev honeycomb model.

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 \(\mathbb{C}\)).

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.

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).

An \([[n,k,d;e]]_q\) EA Galois-qudit stabilizer code for which \(e = n-k\).

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.

A type of \(q\)-ary code whose parameters satisfy the Singleton bound with equality.

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*{(7)}\end{align} becomes an equality.

An \([n\times m,k,d]_q\) rank-metric code whose parameters are such that the sum-rank-metric Singleton bound [651; Prop. 34] \begin{align} d_{\text{SR}}(C) \leq n - k + 1 \tag*{(8)}\end{align} becomes an equality, where \(d_{\text{SR}}\) is the sum-rank metric.

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.

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 [701].

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.

An RGC that corresponds to an extreme point in the storage-bandwidth trade-off curve that is characterised by \(\alpha = d\beta\).

An RGC that corresponds to an extreme point in the storage-bandwidth trade-off curve that is characterised by \(\alpha = (d-k+1)\beta\).

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.

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 [175]. 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 \(q\).

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 analogue of the GKP code. Encodes a qudit into a larger qudit and protects against Pauli shifts up to some maximum value.

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.

A code based on a modular-qudit cluster state.

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.

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 [23] or constructing a star-bipartition; see [707; 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.

Dynamically generated stabilizer-based modular-qudit code whose (not necessarily periodic) sequence of few-body measurements implements state initialization, logical gates and error detection.

A modular-qudit extension of the honeycomb Floquet code.

Monolithic code encoding a qubit into a single modular qudit and protecting against either \(Z\)-type or \(X\)-type modular-qudit Pauli shifts.

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.

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 [707; Sec. VII].

Extension of the surface code to prime-dimensional [36,441] and more general modular qudits [711]. 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.

A sphere packing mapped into a time-dependent electromagnetic signal [712,713]. There is a close relation between abstract real-space encodings and modulation schemes, and certain simple sphere packings are often synonymous with their corresponding modulation schemes.

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 \(K\).

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 [717]. 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 [718,719].

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.

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.

Extension of binary group-orbit codes to multi-antenna transmission.

Irregular LDPC code whose construction generalizes those of the original examples of irregular LDPC as well as RA codes.

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).

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 [724]. Univariate (\(m=1\)) [724,725] and multivariante (\(m>1\)) [726] codes have been proposed.

Multi-mode GKP code whose underlying lattice is utilized in variations of the NTRU cryptosystem [728]. Randomized constructions yield constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability.

A nonlinear \((12,32,5)\) binary code that is the largest double-error-correcting code.

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 \(GF(q)\).

A type of binary code whose parameters satisfy the Johnson bound with equality.

An approximate qubit code obtained from a numerical optimization involving a reinforcement learning agent.

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 [739].

A poset code based on the total ordering of \([n]\), i.e., \(1\leq 2\leq \cdots \leq n\).

One of the 24 positive-definite even unimodular lattices of rank 24.

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 \(q\)-ary code constructed by evaluating functions on an algebraic curve.

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.

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.

Bosonic rotation code consisting of superpositions of Pegg-Barnett phase states [753].

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 [128,129], semidefinite-program recovery/encoding optimization [754,755], or reinforcement learning [756,757].

One of several four-qubit codes that can (approximately) correct a single AD error with higher fidelity than the \([[4,1,2]]\) subcodes of the \([[4,2,2]]\) code.

The unique self-dual linear \((8,4^4)_{\mathbb{Z}_4}\) code of Lee distance 6 and Euclidean distance 8.

Coherent-state c-q binary code whose encoding is either in the vacuum \(|0\rangle\) or in a nonzero coherent state \(|\alpha\rangle\).

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.

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 [764–768].

A length-\(n\) ternary code over \(\{\pm 1,0\}\) whose whose generator matrix has columns with one \(+1\), one \(-1\), and with the rest of the entries zero.

An OAQECC family that encompasses ordinary (i.e., subspace) qubit codes, subsystem qubit codes, and hybrid qubit codes using a unified operator-algebraic framework.

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 \(\mathsf{P}_n\).

A code family that encompasses ordinary (i.e., subspace) codes, subsystem codes, classical-quantum codes, and hybrid codes using a unified operator-algebraic framework.

An LRC whose parameters saturate a generalized Singleton bound.

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.

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.

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.

Encodes \(k\) bosonic modes into \(n\) bosonic modes.

A constant-excitation PI Fock-state code whose construction is based on integer partitions.

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 [790; pg. 107][78; pg. 192] for further details.

Block quantum code defined on two-dimensional subsystems such that any permutation of the subsystems leaves any codeword invariant.

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 \(0\leftrightarrow\alpha\).

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) [793].

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.

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 [795].

A \(t\)-erasure LRC whose coordinate erasures can be recovered in parallel.

A matrix-based code constructed out of two linear binary or \(q\)-ary codes with parity-check matrices \(H_A,H_B\) such its parity-check matrix is \(H_A \otimes H_B\). Its dual has parity-check matrix \(G_A\otimes G_B\), where \(G_{A,B}\) are the generator matrices of the two underlying codes [179].

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.

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 [799] and potentially a dF/CFT duality [800]. It has been generalized to higher dimensions [801] and to include gauge-like degrees of freedom on the links of the tensor network [802,803]. All boundary global symmetries must be dual to bulk gauge symmetries, and vice versa [804].

Non-uniform QSC whose projection is onto a copy of an irreducible representation of the single-qubit Pauli group, the symmetry group of the \(\{2,2,4\}\) tesselation of the sphere. Each codeword is a quantum superposition of vertices of a tetrahedron with \(\pm 1\) coefficients.

Encodes quantum information into superpositions of rotated and translated versions of different Penrose tilings of \(\mathbb{R}^n\).

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 \(21132\).

Spherical \((3,32,(9-\sqrt{5})/6)\) code whose codewords are the vertices of the pentakis dodecahedron, the convex hull of the icosahedron and dodecahedron.

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*{(9)}\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.

A type of \(q\)-ary code whose parameters satisfy the Hamming bound with equality.

A type of block quantum code whose parameters satisfy the quantum Hamming bound with equality.

Block quantum code encoding one subsystem into an odd number \(n\) subsystems whose encoding isometry is a perfect tensor. This code stems from an AME\((n,q)\) AME state, or equivalently, a \(((n+1,1,\lfloor (n+1)/2 \rfloor + 1))\) code.

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.

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 \(S_n\).

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 [811; Appx. A.2.1].

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 [812].

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, \(\mathbb{R}^2\).

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 [814].

Evaluation AG code of bivariate polynomials of some finite maximum degree, evaluated at points lying on an affine or projective plane curve.

Classical code defined on a two-dimensional square lattice whose parity checks are applied on the four vertices of each square.

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 code adapted to transmit classical information over channels with classical inputs and quantum outputs.

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 \(x=u G^{\otimes n}\).

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 (\(q=6\)).

A polytope code in three dimensions, i.e., a spherical code whose codewords form vertices of a polyhedron.

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).

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 in a generalization of the antipodal mapping.

Spherical code whose codewords are the vertices of a polytope, i.e., a geometrical figure bounded by lines, planes, and hyperplanes in either real [3] or complex [836] space. A polytope in two (three, four) dimensions is called a polygon (polyhedron, polychoron).

Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\), with its distance evaluated in the poset metric.

PI qubit code whose recovery succeeds at protecting against AD errors with a success probability less than one.

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.

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 [840,841].

Prime-qudit CSS code constructed using two RS codes.

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 nonzero entries in self-orhogonal rows to \(X\) operators, and \(Z\) operators for each row in the orthogonal complement [843,844].

BCH codes for \(b=1\) and for \(n=q^r-1\) for some \(r\geq 2\).

A code used to obtain information from several servers privately, i.e., without the servers knowing what information was obtained.

Code constructed using two explicit constructions, with each construction corresponding to one of the two extreme points of the storage-bandwidth trade-off curve [848].

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.

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.

A projective code whose codewords all have one of two possible nonzero Hamming weights.

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.

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.

Any one of 13 quaternary extremal Type II self-dual linear codes over \(\mathbb{Z}_4\) of length 24 whose mod-two reduction (mapping \(0,1,2,3\) to \(0,1,0,1\)) is the Golay code [857; Thm. 11]. Each code has Lee distance 12, Hamming distance 8, and Euclidean distance 16 [857; Thm. 9].

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.

An analog code encoding into \(q\) different signals such that each codeword corresponds to a signal.

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 [858].

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 [193; Sec. 31.3.1.1] for an example.

Member of a quadruple of cyclic \(q\)-ary codes of prime length \(n\) where \(q\) is prime and a quadratic-residue modulo \(n\). The codes are constructed using quadratic residues and nonresidues of \(n\). Extensions to prime-power \(q\) are also known [860,861].

A quaternary encoding into a constellation of four points distributed equidistantly on a circle. For the case of \(\pi/4\)-QPSK, the constellation is \(\{e^{\pm i\frac{\pi}{4}},e^{\pm i\frac{3\pi}{4}}\}\).

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 [713; Ch. 16].

Evaluation code of polynomials evaluated on points lying on a quadric hypersurface.

A Galois-qudit CSS code constructed using two linear AG codes.

A Galois-qudit stabilizer code over \(n\) Galois qudits of dimension \(q = 2^n \) that is useful in protecting against faults in qubit Clifford circuits acting on stacked quantum memories. This code can be treated as a code on an \(n\times n\) qubit stacked memory by decomposing each Galois qudit into a Kronecker product of \(n\) qubits; see [227,439,869,870][868; Sec. 5.3].

A Galois-qudit CSS code constructed using two Goppa codes.

Member of a family of \([[n,k,d]]\) 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 \(n\to\infty\). The code can be denoted by \([[n,k,d,w]]\) or, for \(q\)-dimensional Galois-qudit QLDPC codes, \([[n,k,d]]_{(w,q)}\). 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 \(\leq w\). For finite size codes, the notation 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.

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.

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 \(r\) [875; Thm. 2].

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 [461].

CSS code constructed from an extension of check product (between two classical codes) to a product between a classical and a 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.

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.

Stabilizer code designed to correct both data qubit errors and syndrome measurement errors simultaneously due to extra redundancy in its stabilizer generators.

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.

True Galois-qudit stabilizer code constructed from \(q\)-ary duadic codes via the Hermitian construction or the Galois-qudit CSS construction.

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.

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.

CSS code constructed from a hypergraph product of bipartite expander graphs [345] 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 convolutional code whose stabilizer group consists of different constant-size subsets.

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.

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.

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 \(n\).

Member of a family of \([[n,k,d]]\) stabilizer codes for which the number of sites participating in each stabilizer generator is bounded by a constant as \(n\to\infty\).

A type of block quantum code whose parameters satisfy the quantum Singleton bound with equality.

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.

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.

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.

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 [905] using an encoding based on classical polar codes. Variants of the initial scheme have been developed for degradable channels [906] and extended to arbitrary channels [907].

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 [440; Thm. 40] and \(d \geq \sqrt{n}\) when \(n\equiv 1\) modulo 4 [440; Thm. 41].

A CSS code whose qubits are associated with vertices of a simplex graph with \(m+1\) colors.

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 \(|\phi_1\rangle = |-\rangle\).

QLDPC code whose stabilizer generator matrix resembles the parity-check matrix of SC-LDPC codes. There exist CSS [910] and stabilizer constructions [911]. 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.

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.

A qubit stabilizer code designed to protect against synchronization errors (a.k.a. misalignment), which are errors that misalign the code block in a larger block by one or more locations.

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.

A quantum version of the turbo code, obtained from an interleaved serial quantum concatenation [916; Def. 30] of quantum convolutional codes.

Hermitian code arising constructed from twisted BCH codes.

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).

A block code of length \(n\) whose construction was inspired by a quantum 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.

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 [923]. Their simple structure makes them useful for several wireless communication standards.

A QLDPC code such that cyclic shifts of the subsystems by \(\ell\geq 1\) leave the codespace invariant. Such codes have circulant stabilizer generator matrices [924,925].

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. Code for which \(\ell = 1\) are cyclic, while codes for which \(\ell = 2\) are called double circulant.

A block code on \(n\) subsystems such that cyclic shifts of the subsystems by \(\ell\geq 1\) leave the codespace invariant.

An extension of the color code construction to quasi-hyperbolic manifolds, e.g., a product of a 2D hyperbolic surface and a circle.

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.

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.

An extended quadratic residue quaternary linear code over \(\mathbb{Z}_4\) that is a quaternary version of the Golay code. The code has Lee distance 12, Hamming distance 8, and Euclidean distance 16 [928]. The code maps to a binary code under the Gray map whose distance distribution is equal to the MacWilliams transform of the distance distribution of its dual code.

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\((r,m)\) [929; Thm. 19].

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.

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.

Qubit code designed for transmission of classical information in the form of bits through non-classical channels.

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.

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.

A generalization of the 3D surface code to modular qudits.

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.

Generalization of the X-cube model code to modular qudits.

Generalization of the Haah cubic code to modular qudits.

Encodes \(K\)-dimensional Hilbert space into \(n\) bosonic modes.

A subcode of a two-mode GKP code whose projection is onto a copy of an irreducible representation of the single-qutrit Pauli group, the symmetry group of the \(\{3,3,3\}\) tesselation of real space [3; pg. 136]. The code admits the corresponding gates via displacements and phase-space rotations.

An NRT analogue of an RS code.

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.

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).

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.

Code whose encoding is naturally constructed by randomly sampling from a large set of (not necessarily unitary) quantum circuits.

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 \([n\times m,k,d]_q\).

A family of codes in permutations derived from \(q\)-ary linear codes, such as Lee-metric codes, RS codes [941], quadratic-residue codes, and most binary codes.

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.

A three-dimensional cluster-state code defined on the bcc lattice (i.e., a cubic lattice with qubits on edges and faces).

A block code encoding a \(k\)-dimensional vector of real or complex numbers into an \(n\)-dimensional real or complex vector space.

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 [186], 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 \(\{4,8,16\}\).

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 [836; pg. 127][241; pg. 126] for realizations of the 72 codewords.

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.

An \([n,k,n-k+1]_q\) linear code based on polynomials over \(GF(q)\).

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.

An LDPC code whose parity-check matrix has a fixed number of entries for each row or column.

A binary Tanner code defined on a regular bipartite graph, with the inner code being the same for all vertices.

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.

An LDPC code whose parity-check matrix has weight-two columns arranged in a step-like pattern for its last columns [956].

Generalization of the RA code in which two accumulators and permutations are used.

\([n,1,n]\) 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 \(S_n\).

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 \(\omega\).

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.

Encodes \(K\) states (codewords) in \(n\) coordinates over a finite ring \(R\).

A lattice that is symmetric under a specific crystallographic reflection group; see [241; 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.

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.

Member of a \(q\)-ary linear code family that includes many examples of MDS codes that are not GRS codes.

GKP code protecting against small angular position and momentum shifts of a planar rotor.

Rotor analogue of the qubit and analog cluster-state codes. The exact rotor cluster state is non-normalizable, so approximate constructions have to be considered. Defined from an real-valued weighed adjacency matrix of a graph [71].

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 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.

An MDS array code protecting against double erasures.

Evaluation code of polynomials evaluated on points lying on a ruled surface.

Approximate \(n\)-fermionic code whose codewords are low-energy states of the Sachdev-Ye-Kitaev (SYK) Hamiltonian [968,969] or other low-rank SYK models [970,971].

Member of a family of non-CSS subsystem codes constructed from hypergraphs that satisfy certain assumptions [459; Construction C].

Evaluation code of polynomials evaluated on points lying on a Schubert variety.

Evaluation code of polynomials evaluated on points lying on a Segre variety.

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 [976].

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 [375,977], also required finite-spin Hamiltonians.

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.

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.

An additive linear code \(C\) over \(\mathbb{Z}_q\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to the standard inner product.

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.

Member of a family that is related to affine resolvable block designs and that is universally optimal.

A \(t\)-erasure LRC whose coordinate erasures are recovered in sequential fashion.

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 [154].

A fractal type-I fracton CSS code defined on a cubic lattice [21; Eq. (D22)]. The code admits an excitation-moving operator shaped like a Sierpinski triangle [21; Fig. 2].

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 [121; Sec. 7.7] for a parameterization.

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 \(S_n\).

Encodes \(K\)-dimensional Hilbert space into a single bosonic mode. A trivial single-mode code encoding a qubit into the first two Fock states \(\{|0\rangle,|1\rangle\}\) is called the single-rail encoding [983,984].

Block quantum qudit code whose error-syndrome weights increase linearly with the distance of the error state to the code space.

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.

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 [371,987]. Sampling a description of this code can be done with an efficient randomized algorithm with \(2^{-\Omega(n)}\) failure probability.

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.

A member of a family of Galois-qudit CSS codes constructed from skew-cyclic classical codes over rings [989; Thm. 5.8]. See related study [990] that uses cyclic codes over rings.

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 \(R\).

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.

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.

A block code of length \(n\) that either detects or corrects errors on at most two coordinates, i.e., has distance \(d \leq 5\).

A block quantum code on \(n\) subsystems that either detects or corrects errors on at most two subsystems, i.e., have distance \(\leq 5\).

A qubit stabilizer code that either detects or corrects errors on at most two subsystems, i.e., have distance \(\leq 5\).

A \((10,40,1/6)\) spherical code found by W. D. Smith [995] and conjectured to be optimal in terms of minimizing potential energy functions [996]. A beautified set of coordinates can be found on the site [995].

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 [998].

Spherical \((3,24,0.55384)\) code whose codewords are the vertices of the snub cube.

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.

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.

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.

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 \(n\to\infty\).

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 [1008].

An analog code whose points can be thought of as forming centers of (real or complex) spheres that pack (real or complex) space. Sphere packings provide ways of encoding digital or analog information into the frequency, amplitude, and phase of one or more analog waveforms for transmission through, e.g., an optical fiber or free space. This is due to Kotelnikov's [1009] and Shannon's [1010] fundamental observation that a discretized electromagnetic signal of finite bandwidth and average power \(P\) can be represented as a vector in \(\mathbb{R}^n\) with norm \(nP\). Questions of capacity of electromagnetic communication channels then translate to packing problems in \(\mathbb{R}^n\) [241].

Code whose codewords are points on an \(n\)-dimensional sphere \(S^{n}\) with radius one.

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.

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 [1014; Table 1].

An analogue of the single-mode GKP code designed for atomic ensembles. Was designed by using the Holstein-Primakoff mapping [1016] (see also [1017]) 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.

An analogue of the two-component cat code designed for a large spin, which is often realized in the PI subspace of atomic ensembles.

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.

Homological product code whose underlying quantum-code boundary operators are square matrices (see Qubit CSS-to-homology correspondence).

Spherical \((3,8,4(4-\sqrt{2})/7)\) code whose codewords are the vertices of the square antiprism.

Single-mode GKP qudit-into-oscillator CSS 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.

A code based on the cluster state on a square lattice that was used in the the first proposal for MBQC [189,190].

2D 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].

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 \(\pm\alpha\).

Approximate bosonic code that encodes a qubit into the same Fock state, but one which is squeezed in opposite directions.

A special case of a generalized Srivastava code for \(z_j = \alpha_j^{\mu}\) for some \(\mu\) and \(t=1\).

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.

An MDS array code protecting against triple erasures.

A non-CSS color code on a lattice patch with a single twist defect at the center of the patch.

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) [1029].

A code that is a set of subspaces of \(GF(q)^n\).

A \(q\)-ary code that can be mapped into a subspace \(t\)-\((n,w,\lambda)_q\) design.

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.

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 QECC encoding into a \(q^n\)-dimensional Hilbert space consisting of \(n\) Galois qudits.

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.

Family of quantum LDPC codes obtained by combining the subsystem hypgergraph product code construction with classical simplex codes. The result are CSS subsystem codes with weight-three gauge generators and code parameters \([n=(2^r − 1)^2, k=r^2, d=2^{r-1}]\) for \(r \geq 3\).

A quantum code which encodes quantum information in a tensor factor of a subspace that is decomposed into a tensor product of subsystems.

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.

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 [1039].

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.

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 [1041; Sec. III]. The code can be obtained from a generalized Shor code by removing certain stabilizers that do no affect the code distance.

Member of a family of subsystem CSS codes constructed from a subsystem hypergraph product of a lifted binary linear 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 QECC encoding into a \(q^n\)-dimensional Hilbert space consisting of \(n\) modular qudits.

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 QECC encoding into a \(2^n\)-dimensional (i.e., \(n\)-qubit) Hilbert space.

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 version of the rotated surface code.

Subsystem stabilizer code obtained from a spacetime circuit code by gauging out logical operators that correspond to circuit faults with trivial effect [1004; Sec. 5.4].

Subsystem version of the surface code defined on a square lattice with qubits placed at every vertex and center of everry edge.

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.

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.

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).

Evaluation AG code of rational functions evaluated on points lying on a Suzuki curve.

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.

A code whose codewords form the ground-state or low-energy subspace of a code Hamiltonian realizing symmetry-protected topological (SPT) order.

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 \(n\to\infty\) [1057].

A family of \(q\)-ary polynomial evaluation codes that are optimal LRCs and for which \(q\) is comparable to \(n\).