Here is a list of good QLDPC and related codes.

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Code Description Relation
Dinur-Hsieh-Lin-Vidick (DHLV) code 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. DHLV code construction yields asymptotically good QLDPC codes.
Expander LP code Family of \(G\)-lifted product codes constructed using two random classical Tanner codes defined on expander graphs [1]. 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. Lifted products of certain classical Tanner codes are the first asymptotically good QLDPC codes.
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 \(D\). 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. Chain complexes describing some QLDPC codes [2,3], and, more generally, CSS codes [4] can be ‘lifted’ into higher-dimensional manifolds admitting some notion of geometric locality. Applying this procedure to good QLDPC codes yields \([[n,n^{1-2/D},n^{1-1/D}]]\) lattice stabilizer codes in \(D\) spatial dimensions that saturate the BPT bound [3,5,6].
Lattice subsystem code A geometrically local qubit, modular-qudit, or Galois-qudit subsystem stabilizer code with qudits organized on a lattice modeled by the additive group \(\mathbb{Z}^D\) for spatial dimension \(D\). On an infinite lattice, its gauge group is generated by few-site Pauli operators and their translations, in which case the code is called translationally invariant subsystem code. The stabilizer group may contain generators of unbounded weight, distinguishing these codes from stabilizer codes with bounded-weight generators for which some logical qubits were re-assigned to be gauge qubits. 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. An \([[n,k,d]]\) qubit stabilizer code can be converted into an order \([[O(\ell \delta n),k,\Omega(d/w)]]\) subsystem qubit stabilizer code with weight-three gauge operators via the wire-code mapping [7], which uses weight reduction. Here, \(w\) and \(\delta\) are the weight and degree of the input code’s Tanner graph, while \(\ell\) is the length of the longest edge of a particular embedding of that graph. Applying this procedure to good QLDPC codes and using an embedding into \(D\)-dimensional Euclidean space yields lattice subsystem codes whose logical-qubit number and distance both scale as \(\Theta(n^{1-1/D})\) as functions of block length \(n\), saturating the subsystem BT bound [7].
Layer code Member of a family of qubit QLDPC CSS codes with stabilizer generator weights \(\leq 6\) that are obtained by coupling layers of 2D surface codes according to the Tanner graph of a QLDPC code (or a more general qubit stabilizer 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. Layer code parameters, of order \((10,40,4)\), achieve the BPT bound in 3D when asymptotically good QLDPC codes are used in the construction.
Lossless expander balanced-product code 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 [8] yields a \(c^3\)-LTC [9]. Using two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [10]. Taking a balanced product of two-sided expanders, which are only conjectured to exist, yields an asymptotically good QLDPC code family [10].
Quantum Tanner code 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. See Ref. [11] for explicit instances based on dihedral groups. This construction has been generalized to Schreier graphs [12]. Quantum Tanner code construction yields asymptotically good QLDPC codes.
Quantum maximum-distance-separable (MDS) code A type of block quantum code whose parameters satisfy the quantum Singleton bound with equality. AEL distance amplification [13,14] can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound [15; Corr. 5.3].

References

[1]
S. Hoory, N. Linial, and A. Wigderson, “Expander graphs and their applications”, Bulletin of the American Mathematical Society 43, 439 (2006) DOI
[2]
M. Freedman and M. B. Hastings, “Building manifolds from quantum codes”, (2021) arXiv:2012.02249
[3]
T.-C. Lin, A. Wills, and M.-H. Hsieh, “Geometrically Local Quantum and Classical Codes from Subdivision”, (2024) arXiv:2309.16104
[4]
V. Guemard, “Lifts of quantum CSS codes”, (2024) arXiv:2404.16736
[5]
E. Portnoy, “Local Quantum Codes from Subdivided Manifolds”, (2023) arXiv:2303.06755
[6]
X. Li, T.-C. Lin, and M.-H. Hsieh, “Transform Arbitrary Good Quantum LDPC Codes into Good Geometrically Local Codes in Any Dimension”, (2024) arXiv:2408.01769
[7]
N. Baspin and D. Williamson, “Wire Codes”, (2024) arXiv:2410.10194
[8]
M. Capalbo, O. Reingold, S. Vadhan, and A. Wigderson, “Randomness conductors and constant-degree lossless expanders”, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing 659 (2002) DOI
[9]
T.-C. Lin and M.-H. Hsieh, “\(c^3\)-Locally Testable Codes from Lossless Expanders”, (2022) arXiv:2201.11369
[10]
T.-C. Lin and M.-H. Hsieh, “Good quantum LDPC codes with linear time decoder from lossless expanders”, (2022) arXiv:2203.03581
[11]
R. K. Radebold, S. D. Bartlett, and A. C. Doherty, “Explicit Instances of Quantum Tanner Codes”, (2025) arXiv:2508.05095
[12]
O. Å. Mostad, E. Rosnes, and H.-Y. Lin, “Generalizing Quantum Tanner Codes”, (2024) arXiv:2405.07980
[13]
N. Alon, J. Edmonds, and M. Luby, “Linear time erasure codes with nearly optimal recovery”, Proceedings of IEEE 36th Annual Foundations of Computer Science 512 DOI
[14]
N. Alon and M. Luby, “A linear time erasure-resilient code with nearly optimal recovery”, IEEE Transactions on Information Theory 42, 1732 (1996) DOI
[15]
T. Bergamaschi, L. Golowich, and S. Gunn, “Approaching the Quantum Singleton Bound with Approximate Error Correction”, (2022) arXiv:2212.09935