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.
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 (\(q=2\)).

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