Here is a list of code families which contain quantum 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].
Five-qubit perfect code Five-qubit stabilizer code with generators that are symmetric under cyclic permutation of qubits, \begin{align} \begin{split} S_1 &= IXZZX \\ S_2 &= XZZXI \\ S_3 &= ZZXIX \\ S_4 &= ZXIXZ~. \end{split} \end{align} 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 [2][3][4] or the Galois-qudit CSS construction [5][6]. Some Galois-qudit GRS codes are quantum MDS [2].
Galois-qudit RS code Also called a polynomial code (QPyC). An \([[n,k,n-k+1]]_{GF(q)}\) (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 \(n-k_1=k_1-k_2\).
Maximum distance separable (MDS) code A \([n,k,d]_q\) \(q\)-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 \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 Thm. 1.9.10 in Ref. [7]. A code is near MDS (NMDS) if the code and its dual are mode AMDS.
Quantum Reed-Solomon code Also called prime-qudit polynomial code (QPyC). Prime-qudit CSS code constructed using two Reed-Solomon codes. A polynomial code is a quantum MDS code when \(n-k_1=k_1-k_2\).
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 \end{align} becomes an equality.
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 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} \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.
True Galois-qudit stabilizer code Also called a linear stabilizer code. A \([[n,k,d]]_{GF(q)}\) stabilizer code whose stabilizer's symplectic representation forms a linear subspace. In other words, the set of \(q\)-ary vectors representing the stabilizer group is closed under both addition and multiplication by elements of \(GF(q)\). In contrast, Galois-qudit stabilizer codes admit sets of vectors that are closed under addition only. Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [8][9][10][11], in particular from cyclic [1], constacyclic [12][13] and negacyclic [14] codes.
\([[2m,2m-2,2]]\) error-detecting code Also known as the iceberg code. CSS stabilizer code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits. Admits a basis such that each codeword is a superposition of a computational basis state labeled by a bitstring \(b\) and a state labeled by the negation of \(b\). Such states generalize the two-qubit Bell states and three-qubit GHz states and are often called cat states. The \([[2m,2m-2,2]]\) error-detecting code is one of the two qubit quantum MDS [8].
\([[4,2,2]]\) CSS code Also known as the \(C_4\) code. Four-qubit CSS stabilizer code with 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} \end{align} This code is the smallest single-qubit error-detecting code. It is also the smallest instance of the toric code, and its various single-qubit subcodes are small planar surface codes.

References

[1]
G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011). DOI
[2]
Lingfei Jin and Chaoping Xing, “A Construction of New Quantum MDS Codes”. 1311.3009
[3]
X. Liu, L. Yu, and H. Liu, “New quantum codes from Hermitian dual-containing codes”, International Journal of Quantum Information 17, 1950006 (2019). DOI
[4]
L. Jin et al., “Application of Classical Hermitian Self-Orthogonal MDS Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 56, 4735 (2010). DOI
[5]
Dorit Aharonov and Michael Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”. quant-ph/9906129
[6]
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). DOI; 0812.4514
[7]
W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI
[8]
M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004). DOI; quant-ph/0312164
[9]
R. Li and Z. Xu, “Construction of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:msub><mml:mrow /><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>quantum codes for odd prime power<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math>”, Physical Review A 82, (2010). DOI; 0906.2509
[10]
Xianmang He, Liqing Xu, and Hao Chen, “New $q$-ary Quantum MDS Codes with Distances Bigger than $\frac{q}{2}$”. 1507.08355
[11]
Liangdong Lu et al., “New Quantum MDS codes constructed from Constacyclic codes”. 1803.07927
[12]
X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014). DOI
[13]
B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015). DOI
[14]
X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013). DOI