Here is a list of codes related to quantum MDS codes or entanglement-assisted MDS codes.
Code Description MDS Detail
Asymmetric quantum code Quantum systems can be roughly characterized by two types of noise, a bit-flip noise that maps canonical basis states into each other, and a phase-flip noise that induces relative phases between superpositions of such basis states. A code cannot protect against both types of noise arbitrarily well, and there is a tradeoff between the two types of protection. An asymmetric quantum code is one that performs much better against one type of noise than the other type. Such codes typically have tunable distances against each noise type and include CSS codes, GKP codes, and QSCs. An asymmetric Singleton bound and linear programming bounds for asymmetric CSS codes have been formulated [1]. Asymmetric MDS codes have been characterized [2].
Constacyclic code A classical code \(C\) of length \(n\) over an alphabet \(R\) is \(\alpha\)-constacyclic (or \(\alpha\)-twisted) if, for each string \(c_1 c_2 \cdots c_n\in C\), the string \(\alpha c_n, c_1, \cdots, c_{n-1} \in C\). A \(-1\)-constacyclic code is called negacyclic. Many quantum MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [3–6], in particular from cyclic [7], constacyclic [6,8,9], and negacyclic [10] codes.
Cyclic linear \(q\)-ary code A \(q\)-ary code of length \(n\) is cyclic if, for each codeword \(c_1 c_2 \cdots c_n\), the cyclically shifted string \(c_n c_1 \cdots c_{n-1}\) is also a codeword. A cyclic code is called primitive when \(n=q^r-1\) for some \(r\geq 2\). A shortened cyclic code is obtained from a cyclic code by taking only codewords with the first \(j\) zero entries, and deleting those zeroes. Quantum MDS codes can be constructed from \(q\)-ary cyclic codes using the Hermitian construction [7].
EA MDS code EA Galois-qudit code whose parameters make the EAQECC Singleton bound (a.k.a. qubit-ebit Singleton bound) [11; Thm. 6] become an equality.
EA qubit 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 \([[n,k;e]]\) 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.
Five-qubit perfect code Five-qubit cyclic stabilizer code that is the smallest qubit stabilizer code to correct a single-qubit error. The five-qubit code is one of the two qubit quantum MDS codes.
Galois-qudit GRS code True \(q\)-Galois-qudit stabilizer code constructed from GRS codes via either the Hermitian construction [12–14] or the Galois-qudit CSS construction [15,16]. Some Galois-qudit GRS codes are quantum MDS [12].
Galois-qudit RS code An \([[n,k,n-k+1]]_q\) (with \(q>n\)) Galois-qudit CSS code constructed using two RS codes over \(GF(q)\). A polynomial code is a quantum MDS code when \(n-k_1=k_1-k_2\).
Galois-qudit quantum RM code 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 [17; Sec. 4.2]. There exists a quantum RM code \([[q, q − 2ν − 2, ν + 2]]_q\) for \( 0\leq v \leq \frac{(q-2)}{2}\) and \([[q^2,q^2-2v-2,v+2]]_q\) for \(0\leq v \leq q-2\). Both these codes satisfy the quantum Singleton bound.
Generalized RS (GRS) code An \([n,k,n-k+1]_q\) linear code that is a modification of the RS code where codeword polynomials are multiplied by additional prefactors. Some quantum MDS codes are constructed from cyclic and constacyclic codes [18] which are GRS codes [19,20].
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. AEL distance amplification [21,22] can be used to construct asymptotically good QLDPC codes that approach the quantum Singleton bound [23; Corr. 5.3].
Hermitian Galois-qudit code An \([[n,k,d]]_q\) true Galois-qudit stabilizer code constructed from a Hermitian self-orthogonal linear code over \(GF(q^2)\) using the one-to-one correspondence between the Galois-qudit Pauli matrices and elements of the Galois field \(GF(q^2)\). Many quantum MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [3–6], in particular from cyclic [7], constacyclic [6,8,9] and negacyclic [10] codes.
Maximum distance separable (MDS) code A type of \(q\)-ary code whose parameters satisfy the Singleton bound with equality.
Perfect-tensor code Block quantum code encoding one subsystem into \(n\) subsystems whose encoding isometry is a perfect tensor. This code stems from an AME\((n,q)\) AME state, or equivalently, a \(((n+1,1,\lfloor (n+1)/2 \rfloor + 1))_{\mathbb{Z}_q}\) code. AME states for even \(n\) are examples of quantum MDS codes with no logical qubits [24,25]. A family of conjectured perfect-tensor codes is quantum MDS [3].
Quantum data-syndrome (QDS) code Stabilizer code designed to correct both data qubit errors and syndrome measurement errors simultaneously due to extra redundancy in its stabilizer generators. The quantum Singleton bound can be extended to QDS codes [26].
Quantum maximum-distance-separable (MDS) code A type of block quantum code whose parameters satisfy the quantum Singleton bound with equality.
Quantum quadratic-residue (QR) code 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 [27; Thm. 40] and \(d \geq \sqrt{n}\) when \(n\equiv 1\) modulo 4 [27; Thm. 41]. Almost all quantum QR codes for prime-dimensional qudits are quantum MDS [28; Corr. 11].
Singleton-bound approaching AQECC 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 [23,29]. Sampling a description of this code can be done with an efficient randomized algorithm with \(2^{-\Omega(n)}\) failure probability. Singleton-bound approaching AQECCs saturate the quantum Singleton bound.
Skew-cyclic CSS code A member of a family of Galois-qudit CSS codes constructed from skew-cyclic classical codes over rings [30; Thm. 5.8]. See related study [31] that uses cyclic codes over rings. Some quantum MDS codes are constructed from cyclic and constacyclic codes using the Galois-qudit CSS construction [30].
Subsystem Galois-qudit stabilizer code Galois-qudit generalization of a subsystem qubit stabilizer code. Can be obtained by taking a Galois-qudit stabilizer code and assigning some of its logical qubits to be gauge qubits. All pure MDS subsystem stabilizer codes are derived from MDS stabilizer codes [32].
Three-qutrit code A \([[3,1,2]]_3\) prime-qudit CSS code that is the smallest qutrit stabilizer code to detect a single-qutrit error. with stabilizer generators \(ZZZ\) and \(XXX\). The code defines a quantum secret-sharing scheme and serves as a minimal model for the AdS/CFT holographic duality. It is also the smallest non-trivial instance of a quantum maximum distance separable code (QMDS), saturating the quantum Singleton bound. The three-qutrit code is the smallest nontrivial quantum MDS code.
\([[2m,2m-2,2]]\) error-detecting code Self-complementary CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. The code is constructed via the CSS construction from an SPC code and a repetition code [33; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [34]. The \([[2m,2m-2,2]]\) error-detecting code forms one of the two families qubit quantum MDS codes [3].
\([[4,2,2]]\) Four-qubit code Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error.
\([[6,2,3]]_{q}\) code Six-qudit MDS error-detecting code defined for Galois-qudit dimension \(q=3\) [35], \(q=2^2\) [36], and \(q \geq 5\) [27,35]. This code cannot exist for qubits (\(q=2\)).
\([[6,4,2]]\) error-detecting code Error-detecting six-qubit code with rate \(2/3\) whose codewords are cat/GHZ states. A set of stabilizer generators is \(XXXXXX\) and \(ZZZZZZ\). It is the unique code for its parameters, up to equivalence [34; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [37].
\([[7,3,3]]_{q}\) code Seven-qudit MDS error-detecting code defined for Galois-qudit dimension \(q=3\) [35] and \(q \geq 7\) [27,35]. This code cannot exist for qubits (\(q=2\)).

References

[1]
P. K. Sarvepalli, A. Klappenecker, and M. Rötteler, “Asymmetric quantum codes: constructions, bounds and performance”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 1645 (2009) DOI
[2]
M. F. EZERMAN, S. JITMAN, H. M. KIAH, and S. LING, “PURE ASYMMETRIC QUANTUM MDS CODES FROM CSS CONSTRUCTION: A COMPLETE CHARACTERIZATION”, International Journal of Quantum Information 11, 1350027 (2013) arXiv:1006.1694 DOI
[3]
M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
[4]
R. Li and Z. Xu, “Construction of[[n,n−4,3]]qquantum codes for odd prime powerq”, Physical Review A 82, (2010) arXiv:0906.2509 DOI
[5]
X. He, L. Xu, and H. Chen, “New \(q\)-ary Quantum MDS Codes with Distances Bigger than \(\frac{q}{2}\)”, (2015) arXiv:1507.08355
[6]
L. Lu, W. Ma, R. Li, Y. Ma, and L. Guo, “New Quantum MDS codes constructed from Constacyclic codes”, (2018) arXiv:1803.07927
[7]
G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011) DOI
[8]
X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014) DOI
[9]
B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015) DOI
[10]
X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013) DOI
[11]
M. Grassl, F. Huber, and A. Winter, “Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes”, IEEE Transactions on Information Theory 68, 3942 (2022) arXiv:2010.07902 DOI
[12]
L. Jin and C. Xing, “A Construction of New Quantum MDS Codes”, (2020) arXiv:1311.3009
[13]
X. Liu, L. Yu, and H. Liu, “New quantum codes from Hermitian dual-containing codes”, International Journal of Quantum Information 17, 1950006 (2019) DOI
[14]
L. Jin, S. Ling, J. Luo, and C. Xing, “Application of Classical Hermitian Self-Orthogonal MDS Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 56, 4735 (2010) DOI
[15]
D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
[16]
Z. Li, L.-J. Xing, and X.-M. Wang, “Quantum generalized Reed-Solomon codes: Unified framework for quantum maximum-distance-separable codes”, Physical Review A 77, (2008) arXiv:0812.4514 DOI
[17]
C.-Y. Lai and C.-C. Lu, “A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices”, IEEE Transactions on Information Theory 57, 7163 (2011) arXiv:0712.0103 DOI
[18]
M. Grassl and M. Rotteler, “Quantum MDS codes over small fields”, 2015 IEEE International Symposium on Information Theory (ISIT) 1104 (2015) arXiv:1502.05267 DOI
[19]
S. Ball, “Grassl–Rötteler cyclic and consta-cyclic MDS codes are generalised Reed–Solomon codes”, Designs, Codes and Cryptography 91, 1685 (2022) DOI
[20]
H. Liu and S. Liu, “A class of constacyclic codes are generalized Reed–Solomon codes”, Designs, Codes and Cryptography 91, 4143 (2023) DOI
[21]
N. Alon, J. Edmonds, and M. Luby, “Linear time erasure codes with nearly optimal recovery”, Proceedings of IEEE 36th Annual Foundations of Computer Science DOI
[22]
N. Alon and M. Luby, “A linear time erasure-resilient code with nearly optimal recovery”, IEEE Transactions on Information Theory 42, 1732 (1996) DOI
[23]
T. Bergamaschi, L. Golowich, and S. Gunn, “Approaching the Quantum Singleton Bound with Approximate Error Correction”, (2022) arXiv:2212.09935
[24]
A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
[25]
Z. Raissi, C. Gogolin, A. Riera, and A. Acín, “Optimal quantum error correcting codes from absolutely maximally entangled states”, Journal of Physics A: Mathematical and Theoretical 51, 075301 (2018) arXiv:1701.03359 DOI
[26]
A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Quantum Data-Syndrome Codes”, IEEE Journal on Selected Areas in Communications 38, 449 (2020) arXiv:1907.01393 DOI
[27]
A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
[28]
E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
[29]
D. Leung and G. Smith, “Communicating over adversarial quantum channels using quantum list codes”, (2007) arXiv:quant-ph/0605086
[30]
H. Q. Dinh, T. Bag, A. K. Upadhyay, R. Bandi, and R. Tansuchat, “A class of skew cyclic codes and application in quantum codes construction”, Discrete Mathematics 344, 112189 (2021) DOI
[31]
M. Ashraf and G. Mohammad, “Quantum codes over Fp from cyclic codes over Fp[u, v]/〈u2 − 1, v3 − v, uv − vu〉”, Cryptography and Communications 11, 325 (2018) DOI
[32]
S. A. Aly and A. Klappenecker, “Subsystem Code Constructions”, (2008) arXiv:0712.4321
[33]
N. Rengaswamy, R. Calderbank, H. D. Pfister, and S. Kadhe, “Synthesis of Logical Clifford Operators via Symplectic Geometry”, 2018 IEEE International Symposium on Information Theory (ISIT) (2018) arXiv:1803.06987 DOI
[34]
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum Error Correction via Codes over GF(4)”, (1997) arXiv:quant-ph/9608006
[35]
Keqin Feng, “Quantum codes [[6, 2, 3]]/sub p/ and [[7, 3, 3]]/sub p/ (p ≥ 3) exist”, IEEE Transactions on Information Theory 48, 2384 (2002) DOI
[36]
Z. Wang, S. Yu, H. Fan, and C. H. Oh, “Quantum error-correcting codes over mixed alphabets”, Physical Review A 88, (2013) arXiv:1205.4253 DOI
[37]
H. Goto, “High-performance fault-tolerant quantum computing with many-hypercube codes”, Science Advances 10, (2024) arXiv:2403.16054 DOI