Here is a list of code families which contain 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 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 code that satisfies the generalization of the quantum Singleton bound to EA codes [11; Thm. 6]. | |

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 generalized Reed-Solomon (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 | Also called a polynomial code (QPyC). An \([[n,k,n-k+1]]_q\) (with \(q>n\)) Galois-qudit CSS code constructed using two Reed-Solomon 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 either the Hermitian construction or the Galois-qudit CSS construction. | 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. Each message \(\mu\) is encoded into a string of values of the corresponding polynomial \(f_\mu\) at the points \(\alpha_i\), multiplied by a corresponding nonzero factor \(v_i \in GF(q)\), \begin{align} \mu\to\left( v_{1}f_{\mu}\left(\alpha_{1}\right),v_{2}f_{\mu}\left(\alpha_{2}\right),\cdots,v_{n}f_{\mu}\left(\alpha_{n}\right)\right)~. \tag*{(1)}\end{align} | Some MDS codes are constructed from cyclic and constacyclic codes [17] which are GRS codes [18,19]. |

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 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 \([n,k,d]_q\) binary or \(q\)-ary linear code is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the Singleton bound \begin{align} d \leq n-k+1 \tag*{(2)}\end{align} becomes an equality. A code is called almost MDS (AMDS) when \(d=n-k\). A bound for general \(q\)-ary codes can also be formulated; see [20; Thm. 1.9.10]. A code is near MDS (NMDS) if the code and its dual are mode AMDS. | |

Quantum maximum-distance-separable (MDS) code | An \(((n,K,d))\) code constructed out of \(q\)-dimensional qubits or Galois qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound is satisfied. | |

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. 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 [21; Thm. 5.8]. See related study [22] that uses cyclic codes over rings. | Some MDS codes are constructed from cyclic and constacyclic codes using the Galois-qudit CSS construction [21]. |

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 codewords are \begin{align} \begin{split} | \overline{0} \rangle &= \frac{1}{\sqrt{3}} (| 000 \rangle + | 111 \rangle + | 222 \rangle) \\ | \overline{1} \rangle &= \frac{1}{\sqrt{3}} (| 012 \rangle + | 120 \rangle + | 201 \rangle) \\ | \overline{2} \rangle &= \frac{1}{\sqrt{3}} (| 021 \rangle + | 102 \rangle + | 210 \rangle)~. \end{split} \tag*{(3)}\end{align} The elements in the superposition of each logical codeword are related to each other via cyclic permutations. | The three-qutrit code is the smallest nontrivial quantum MDS code. |

\([[2m,2m-2,2]]\) error-detecting code | CSS stabilizer 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 [23; Sec. III]. This is the highest-rate distance-two code when an even number of qubits is used [24]. | The \([[2m,2m-2,2]]\) error-detecting code forms one of the two families qubit quantum MDS codes [3]. |

\([[4,2,2]]\) CSS 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 \(q=3\) [25], \(q=2^2\) [26], and \(q \geq 5\) [25,27]. 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 local equivalence [24; Tab. III]. Concatenations of this code with itself yield the \([[6^r,4^r,2^r]]\) level-\(r\) many-hypercube code [28]. | |

\([[7,3,3]]_{q}\) code | Seven-qudit MDS error-detecting code defined for Galois-qudit dimension \(q=3\) [25] and \(q \geq 7\) [25,27]. This code cannot exist for qubits (\(q=2\)). |

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