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