Here is a list of code families which contain quantum MDS codes or entanglement-assisted MDS codes.
Code Description MDS Detail
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 [1].
EA MDS code EA code that satisfies the generalization of the quantum Singleton bound to EA codes [2; 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 [3–5] or the Galois-qudit CSS construction [6,7]. Some Galois-qudit GRS codes are quantum MDS [3].
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.
Hermitian-construction 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 [8–11], in particular from cyclic [1], constacyclic [12,13] and negacyclic [14] 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*{(1)}\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 [15; 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 \(((n,q^k,d))\) code constructed out of \(q\)-dimensional qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound \begin{align} 2(d-1) \leq n-k \tag*{(2)}\end{align} becomes an equality. Such codes are pure [16].
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.
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. This is the highest-rate distance-two code when an even number of qubits is used [17]. The \([[2m,2m-2,2]]\) error-detecting code is one of the two qubit quantum MDS codes [8].
\([[4,2,2]]\) CSS code Four-qubit CSS stabilizer code is the smallest qubit stabilizer code to detect a single-qubit error. Admits generators \(\{XXXX, ZZZZ\} \) and codewords \begin{align} \begin{split} |\overline{00}\rangle = (|0000\rangle + |1111\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{01}\rangle = (|0011\rangle + |1100\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{10}\rangle = (|0101\rangle + |1010\rangle)/\sqrt{2}~{\phantom{.}}\\ |\overline{11}\rangle = (|0110\rangle + |1001\rangle)/\sqrt{2}~. \end{split} \tag*{(4)}\end{align} This code is the smallest instance of the toric code, and its various single-qubit subcodes are small planar surface codes.

References

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