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 codes is the smallest qubit quantum MDS code.
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]. Some Galois-qudit GRS codes are quantum MDS [2].
Galois-qudit RS code Also called 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. [6]. 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.
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 [7][8][9][10], in particular from cyclic [1], constacyclic [11][12] and negacyclic [13] 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]
W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI
[7]
M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004). DOI; quant-ph/0312164
[8]
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
[9]
Xianmang He, Liqing Xu, and Hao Chen, “New $q$-ary Quantum MDS Codes with Distances Bigger than $\frac{q}{2}$”. 1507.08355
[10]
Liangdong Lu et al., “New Quantum MDS codes constructed from Constacyclic codes”. 1803.07927
[11]
X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014). DOI
[12]
B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015). DOI
[13]
X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013). DOI