Here is a list of code families which contain MDS codes.
Code Description MDS Detail
Extended RS code Stub. If \(f\in \mathcal{P}_k\) with \(k<q\), then \(\sum_{\alpha\in\mathbb{F}_q}f(\alpha)=0\) which implies RS codes are odd-like. Hence, by adding a parity check coordinate with evaluation point \(\alpha_0=0\) to an RS code on \(q-1\) registers, the distance increases to \(\hat{d}=d+1\). This addition yields an \([q,k,q-k+1]\) extended RS code.
Folded RS code Stub.
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. [1].
Maximum-rank distance (MRD) code Also called an optimal rank-distance code. An \([n\times m,k,d]_q\) rank-metric code whose parameters are such that the Singleton-like bound \begin{align} k \leq \max(n, m) (\min(n, m) - d + 1) \end{align} become an equality. MRD codes are matrix-code analogues of MDS codes.
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.
Reed-Solomon (RS) code An \([n,k,n-k+1]_q\) linear code based on polynomials over \(GF(q)\). Let \(\{\alpha_1,\cdots,\alpha_n\}\) be \(n\) distinct nonzero elements of \(GF(q)\) with \(q>n\). An RS code encodes \(\mu=\{\mu_0,\cdots,\mu_{k-1}\}\) into \(\{f_\mu(\alpha_1),\cdots,f_\mu(\alpha_n)\}\), with polynomial \begin{align} f_\mu(x)=\mu_0+\mu_1 x + \cdots + \mu_{k-1}x^{k-1}. \end{align} In other words, each codeword \(\mu\) is a string of values of the corresponding polynomial \(f_\mu\) at the points \(\alpha_i\). Every RS code is MDS. If \(k \leq p\), then all linear MDS codes \( [n,k,n-k+1]_{p^m} \) are RS codes [2].


W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI
S. Ball, “On sets of vectors of a finite vector space in which every subset of basis size is a basis”, Journal of the European Mathematical Society 733 (2012). DOI