Here is a list of code families which contain quantum MDS codes.
Code Description MDS Detail
Galois-qudit polynomial code (QPyC) Also called quantum Reed-Solomon code. An \([[n,k,n-k+1]]_{GF(q)}\) (with \(q>n\)) Galois-qudit CSS code constructed using two Reed-Solomon codes over \(GF(q)\). Let \(C_1\) be a \([n,k_1,d_1]_q\) Reed-Solomon code and \(C_2^\perp\) be a \([n,k_2,d_2]_q\) Reed-Solomon code, modified such that \(C_2^\perp \subseteq C_1\) and \(0\le k_2 \le k_1 \le n\). Then, a polynomial code is a non-degenerate \([[n,k_2,d]]_{GF(q)}\) Galois-qudit CSS code with \(d=\min(n-k_1+1,k_1-k_2+1)\). The polynomial code is the span of the basis codewords over GF(\(q\)) \begin{align} |\overline{\beta_0,\cdots,\beta_{k_2-1}}\rangle = \sum_{(\beta_{k_2},\cdots,\beta_{k_1-1})\in GF(q) } \bigotimes_{i=1}^{n} \left|\sum_{j=0}^{k_1-1} \beta_j \alpha_i^j \right\rangle, \end{align} where \((\alpha_1, \cdots, \alpha_n)\) are \(n\) distinct points chosen for code \(C_1\) from \(GF(q)\setminus \{0\}\). A polynomial code is a quantum MDS code when \(n-k_1=k_1-k_2\).
Generalized Reed-Solomon (GRS) code Stub. Quantum MDS codes can be constructed through classical generalized Reed-Solomon codes [1].
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. [2].
Prime-qudit polynomial code (QPyC) Also called quantum Reed-Solomon code. An \([[n,k,n-k+1]]_p\) (with prime \(p>n\)) prime-qudit CSS code constructed using two Reed-Solomon codes over \(GF(p)=\mathbb{Z}_p\). Let \(\{\alpha_1,\cdots,\alpha_n\}\) be \(n\) distinct nonzero elements of \(\mathbb{Z}_p\), and let \(g\) be a number satisfying \(0\leq k \leq g < n\). Then, define degree-\(g\) polynomials \begin{align} f_{\mu\cup c}\left(x\right)=\mu_{0}+\mu_{1}x+\cdots+\mu_{k-1}x^{k-1}+c_{k}x^{k}+\cdots+c_{g}x^{g}\,, \end{align} where the first \(k\) coefficients are indexed by the coefficient vector \(\mu\in\mathbb{Z}_p^{\times k}\), and the remaining coefficients are indexed by the vector \(c\in\mathbb{Z}_p^{\times (g+1-k)}\). Logical states, labeled by \(\mu\), are superpositions of canonical basis states whose \(i\)th bit is \(f_{\mu\cup c}\), evaluated at \(\alpha_i\) and summed over all possible vectors \(c\), \begin{align} |\overline{\mu}\rangle=\sum_{c\in\mathbb{Z}_{p}^{\times(g+1-k)}}|f_{\mu\cup c}(\alpha_{1}),|f_{\mu\cup c}(\alpha_{2}),\cdots,|f_{\mu\cup c}(\alpha_{n})\rangle. \end{align} 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.
\([[5,1,3]]\) 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 smallest quantum MDS code.

References

[1]
Hualu Liu and Xiusheng Liu, “Constructions of quantum MDS codes”. 2002.06040
[2]
W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI