Here is a list of non-CSS non-qubit stabilizer codes. For qubit non-CSS stabilizer codes, see Qubit stabilizer codes (non-CSS). For CSS codes, see Quantum CSS codes (non-qubit) and Qubit CSS codes.

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Code Description
1D lattice stabilizer code Lattice stabilizer code in one Euclidean dimension, using either the ordinary block notion of locality or the fermionic/Majorana notion of locality.
2D lattice stabilizer code Lattice stabilizer code in two Euclidean dimensions, using either the ordinary block notion of locality or the fermionic/Majorana notion of locality.
3D lattice stabilizer code Lattice stabilizer code in three Euclidean dimensions, using either the ordinary block notion of locality or the fermionic/Majorana notion of locality.
4D lattice stabilizer code Lattice stabilizer code in four Euclidean dimensions, using either the ordinary block notion of locality or the fermionic/Majorana notion of locality.
Abelian TQD stabilizer code Modular-qudit stabilizer code whose codewords realize a 2D Abelian twisted-quantum-double topological order on composite-dimensional qudits. For every finite Abelian group \(G=\prod_i \mathbb{Z}_{N_i}\) and every product of Type-I and Type-II cocycles, there is a Pauli stabilizer Hamiltonian realizing the corresponding Abelian TQD [1]. Equivalently, these codes exhaust the 2D Abelian topological orders that admit gapped boundaries [1,2].
Abelian quantum-double stabilizer code 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. The \(G=\mathbb{Z}_2\) instance on a torus is the toric code, and cyclic-group instances reduce to modular-qudit surface codes. All such codes can be realized by a stack of modular-qudit surface codes because all finite Abelian groups are direct products of cyclic groups.
Analog cluster-state code A code based on a continuous-variable (CV), or analog, cluster state. Such a state can be used to perform MBQC of logical modes, which substitutes the temporal dimension necessary for decoding a conventional code with a spatial dimension. The exact analog cluster state is non-normalizable, so approximate constructions have to be considered.
Analog stabilizer code An oscillator-into-oscillator stabilizer code encoding logical oscillator modes into \(n\) physical modes. If the code is defined by \(r\) independent nullifiers, then it is denoted by \([[n,n-r]]_{\mathbb{R}}\) and encodes \(k=n-r\) logical modes [3]. Any analog stabilizer state can be thought of as a pure Gaussian state that has been infinitely squeezed on all modes [3].
Bosonic stabilizer code Bosonic code whose codespace is defined as the common \(+1\) eigenspace of a group of mutually commuting displacement operators. Displacements form the stabilizers of the code, and have continuous eigenvalues, in contrast with the discrete set of eigenvalues of qubit stabilizers. As a result, exact codewords are non-normalizable, so approximate constructions have to be considered. 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.
Chiral semion Walker-Wang model code 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 [4,5]. The corresponding phase can also be realized via a non-stabilizer Hamiltonian [6].
Concatenated GKP code 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.
Double-semion stabilizer code A 2D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) that realizes the 2D double semion topological phase. The code can be obtained from a \(\mathbb{Z}_4\) toric-code ground state by condensing the emergent boson \(e^2 m^2\); in the stabilizer construction this condensation is implemented by two-body measurements [1,7]. Its ground-state subspace can be mapped to that of the double-semion string-net model by a finite-depth quantum circuit with ancillas [1].
Fracton stabilizer code 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.
Frobenius code A cyclic prime-qudit stabilizer code whose length \(n\) divides \(p^t + 1\) for some positive integer \(t\).
GKP CV-cluster-state code A cluster-state code that utilizes a generalized analog cluster state 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 \(V_{jk}\). The code provides a way to perform fault-tolerant MBQC, with the required number \(n-k\) of GKP-encoded physical modes determined by the particular protocol [8–11].
GKP-surface code A concatenated code whose outer code is a GKP code and whose inner code is a surface code, including toric surface-code variants [10,12], rotated surface codes [13–16], and XZZX surface codes [17].
Galois-qudit BCH code 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 [18], e.g., as in Ref. [19].
Galois-qudit GRS code True \(q\)-Galois-qudit stabilizer code constructed from GRS codes via either the Hermitian construction [20–22] or the Galois-qudit CSS construction [23,24].
Galois-qudit RS code A Galois-qudit CSS code family (with \(q>n\)) constructed using two RS codes over \(\mathbb{F}_q\).
Galois-qudit quantum RM code True Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via the Galois-qudit Hermitian construction, the Galois-qudit CSS construction, or directly from their parity-check matrices [25][26; Sec. 4.2].
Galois-qudit stabilizer code 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.
Generalized homological-product code Stabilizer code whose properties are determined from an underlying chain complex, which often consists of some type of product of other chain complexes. The Qubit CSS-to-homology correspondence yields an interpretation of codes in terms of chain complexes, thus allowing for the use of various products from homology in constructing codes.
Good QLDPC code Also called asymptotically good QLDPC codes. A family of QLDPC codes \([[n_i,k_i,d_i]]\) whose asymptotic rate \(\lim_{i\to\infty} k_i/n_i\) and asymptotic distance \(\lim_{i\to\infty} d_i/n_i\) are both positive.
Gottesman-Kitaev-Preskill (GKP) code 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 \(A\). Any GKP code can be generated from a Gram matrix in standard form via a Gaussian unitary transformation [27; Corr. 1].
Graph quantum code A stabilizer code on tensor products of \(G\)-valued qudits for Abelian \(G\) whose encoding isometry is defined using a graph [28; 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 [28]; see [29; 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 [30].
Hermitian Galois-qudit code An \([[n,k,d]]_q\) true Galois-qudit stabilizer code constructed from a Hermitian self-orthogonal linear code over \(\mathbb{F}_{q^2}\) using the one-to-one correspondence between the Galois-qudit Pauli matrices and elements of the Galois field \(\mathbb{F}_{q^2}\).
Hexagonal GKP code 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 [31; Sec. VI].
Lattice stabilizer code A geometrically local stabilizer code with sites organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\), using either the ordinary block notion of locality or the fermionic/Majorana notion of locality. 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.
Locally compact Abelian (LCA) stabilizer code A mixed oscillator stabilizer code whose codewords are quantum lattice states defined on any number of qudits and a nonzero number of oscillators. Its stabilizers are countably infinite subgroups of the qudit Pauli and oscillator displacement groups. Codewords are entangled across the qudit-oscillator bipartition.
Modular-qudit cluster-state code A code based on a modular-qudit cluster state.
Modular-qudit stabilizer code 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 \(\mathsf{S}\) [32; Sec. 3.6]. 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.
NTRU-GKP code Multi-mode GKP code whose underlying lattice is utilized in variations of the NTRU cryptosystem [33]. Randomized constructions yield constant-rate GKP code families whose largest decodable displacement length scales as \(O(\sqrt{n})\) with high probability.
Oscillator-into-oscillator GKP code Multimode GKP code with an infinite-dimensional logical space. Can be obtained by considering an \(n\)-mode GKP code with a finite-dimensional logical space, removing stabilizers such that the logical space becomes infinite dimensional, and applying a Gaussian circuit.
QLDPC code Member of a family of 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 as \(n\to\infty\). Sometimes, the two parameters are explicitly stated: each site of an \((l,w)\)-regular QLDPC code is acted on by \(\leq l\) generators of weight \(\leq w\).
Quantum AG code True Galois-qudit stabilizer code constructed from evaluation AG codes via the Galois-qudit Hermitian construction or the Galois-qudit CSS construction.
Quantum Gabidulin code 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 [34,36–39][35; Sec. 5.3].
Quantum Hermitian AG code Quantum AG code constructed from Hermitian AG codes via the Galois-qudit Hermitian construction or the Galois-qudit CSS construction. The underlying classical codes can be constructed from one-point [40] or two-point [41] Hermitian curves (see also Ref. [42]). In parameter ranges where two-point Hermitian codes improve on one-point codes, the resulting quantum codes can also have improved parameters [41].
Quantum duadic code True Galois-qudit stabilizer code constructed from \(q\)-ary duadic codes via the Hermitian construction or the Galois-qudit CSS construction. Large subclasses of quantum duadic codes are degenerate [43; Sec. 5.5].
Quantum lattice code Bosonic stabilizer code on \(n\) bosonic modes whose stabilizer group is an infinite countable group of oscillator displacement operators which implement lattice translations in phase space.
Quantum low-weight check (QLWC) code 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 \(n\to\infty\).
Quantum plane-curve code Quantum AG code constructed from plane-curve codes via the Galois-qudit Hermitian construction. Code parameters are \([[q^3,q^3+q^2-3q-2r,r+2q-q^2]]_q\), where \(r\) is an integer satisfying \(q^2 - 2 \leq r \leq q^2 + q - 3\), and where the underlying plane curve is \(y^q + y = x^{q-1}\).
Quantum twisted code Hermitian stabilizer code constructed from twisted BCH codes.
Quasi-cyclic QLDPC (QC-QLDPC) code A QLDPC code such that cyclic shifts of the subsystems by \(\ell\geq 1\) leave the codespace invariant. Stabilizer generator matrices of such codes can be put into block form, where each nonzero block is a circulant matrix [44,45].
Qudit cubic code Generalization of the Haah cubic code to modular qudits.
Random stabilizer code An \(n\)-qubit, modular-qudit, or Galois-qudit stabilizer code whose construction is non-deterministic. Since stabilizer encoders are Clifford circuits, such codes can be thought of as arising from random Clifford circuits.
Rotor cluster-state code 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 a real-valued weighted adjacency matrix of a graph [3].
Rotor stabilizer code Rotor code whose codespace is defined as the common \(+1\) eigenspace of a group of mutually commuting rotor generalized Pauli operators. The stabilizer group can be either discrete or continuous, corresponding to modular or linear constraints on angular positions and momenta. Both cases can yield finite or infinite logical dimension. Exact codewords are non-normalizable, so approximate constructions have to be considered.
Stabilizer code A code whose logical subspace is the joint eigenspace (usually with eigenvalue \(+1\)) of a set of commuting unitary Pauli-type operators forming the code’s stabilizer group. They can be block codes defined on tensor-product spaces of qubits or qudits, or non-block codes defined on single sufficiently large Hilbert spaces such as bosonic modes or other Abelian group spaces.
True Galois-qudit stabilizer code A \([[n,k,d]]_q\) stabilizer code whose stabilizer’s Galois symplectic representation forms a linear subspace. In other words, the set of \(q\)-ary vectors representing the stabilizer group is closed under both addition and multiplication by elements of \(\mathbb{F}_q\). In contrast, Galois-qudit stabilizer codes admit sets of vectors that are closed under addition only.
\(D_4\) hyper-diamond GKP code Two-mode GKP qubit-into-oscillator code based on the \(D_4\) hyper-diamond lattice [46].
\(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons GKP code A non-CSS multimode GKP code defined on a 2D mode lattice that encodes a qudit logical space and whose excitations are characterized by the \(U(1)_{2n} \times U(1)_{-2m}\) Chern-Simons theory. The code can be obtained from the analog surface code by condensing certain anyons [47].
\([[10,1,4]]_{G}\) tenfold code A \([[10,1,4]]_{G}\) group code for finite Abelian \(G\). The code is defined using a graph that is closely related to the \([[5,1,3]]\) code.
\([[5,1,3]]_q\) Galois-qudit code True stabilizer code that generalizes the five-qubit perfect code to Galois qudits of prime-power dimension \(q=p^m\). It has \(4(m-1)\) stabilizer generators expressed as \(X_{\gamma} Z_{\gamma} Z_{-\gamma} X_{-\gamma} I\) and its cyclic permutations, with \(\gamma\) iterating over basis elements of \(\mathbb{F}_q\) over \(\mathbb{F}_p\).
\([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code An analog stabilizer version of the five-qubit perfect code, encoding one mode into five and correcting arbitrary errors on any one mode.
\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code Modular-qudit stabilizer code that generalizes the five-qubit perfect code using properties of the multiplicative group \(\mathbb{Z}_q\) [48]; see also [49; Thm. 13]. It has four stabilizer generators consisting of \(X Z Z^\dagger X^\dagger I\) and its cyclic permutations.
\([[5,1,3]]_{\mathbb{Z}}\) Five-rotor code Extension of the five-qubit stabilizer code to the integer alphabet, i.e., the angular momentum states of a rotor. The code is \(U(1)\)-covariant and ideal codewords are not normalizable.
\([[6,2,3]]_{q}\) code Six-qudit MDS error-detecting code defined for Galois-qudit dimension \(q=3\) [50], \(q=2^2\) [51], and \(q \geq 5\) [50][52; Exam. 33]. This code cannot exist for qubits (\(q=2\)).
\([[7,3,3]]_{q}\) code Seven-qudit MDS error-detecting code defined for Galois-qudit dimension \(q=3\) [50] and \(q \geq 7\) [50][52; Exam. 33]. This code cannot exist for qubits (\(q=2\)).
\([[9,1,5]]_3\) quantum Glynn code Nine-qutrit pure Hermitian code that is the smallest qutrit stabilizer code to correct two-qutrit errors.

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