Here is a list of classical codes related to and generalizing the MDS codes.

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Code Description
B-code Binary MDS block array code \(\mathcal{B}_2(m)\) on \((m-1)\times m\) arrays with horizontal and toroidal diagonal parity checks [1].
Denniston code Projective code that is part of a family of \([2^{a+i}+2^i-2^a,3,2^{a+i}-2^a]_{2^a}\) codes for \(i < a\) constructed using Denniston arcs [2; Sec. 19.7.3].
Diagonal code Member of an explicit family of high-rate \([n,k,d,\alpha, \beta = \frac{\alpha}{d-k+1}, M=k\alpha]\) MSR codes for any \(r\) and \(n\). Such codes can optimally repair any \(f\) failed nodes from any \(d\) helper nodes for all \(d\), \(1 \le f \le r\) and \(k \le d \le n-f\) simultaneously. These codes can be constructed over any base field \(\mathbb{F}_q\) as long as \(|\mathbb{F}_q| \ge sn\), where \(s = \text{lcm}(1,2,\cdots,r)\).
EVENODD code Binary array code \(\mathcal{EO}_2(m)\) with independent horizontal and diagonal parity columns, designed to retain optimal double-erasure protection while simplifying small updates [1].
Generalized RS (GRS) code An \([n,k,n-k+1]_q\) linear code that is a modification of the RS code where codeword polynomials are multiplied by additional prefactors [3; Def. 15.3.19].
Griesmer code A type of \(q\)-ary code whose parameters satisfy the Griesmer bound with equality.
Hirschfeld code A \([q+1,4,q-2]_q\) projective geometry code for non-prime \(q\) that is an example of an MDS code that is not an RS code; see [4; Exam. 7.6] for the generator matrix.
Linearized RS code A code obtained by linearizing a skew RS code, i.e., by translating evaluations of skew polynomials into operator evaluations over blocks.
MDS array code An \((n,k,m)\) array code whose codewords can be recovered by any \(k\) out of \(n\) nodes, where each node stores a length-\(m\) column of the codeword. MDS array codes are MDS codes when each matrix codeword is treated as a vector by converting each column into a single coordinate via subpacketization.
Maximum distance separable (MDS) code A \(q\)-ary linear code whose parameters satisfy the Singleton bound with equality.
Maximum-rank distance (MRD) code A rank-metric code whose parameters satisfy the rank-metric Singleton-like bound with equality.
Maximum-sum-rank distance (MSRD) code A sum-rank-metric code whose parameters satisfy the sum-rank-metric Singleton bound with equality.
Minimum-storage regenerating (MSR) code A regenerating code that corresponds to the minimum-storage extreme point of the storage-bandwidth trade-off curve, characterized by \(\alpha = (d-k+1)\beta\).
Narrow-sense RS code An \([q-1,k,n-k+1]_q\) RS code whose points \(\alpha_i\) are all \((i-1)\)st powers of a primitive element \(\alpha\) of \(\mathbb{F}_q\).
Reed-Solomon (RS) code An \([n,k,n-k+1]_q\) linear code based on polynomials over \(\mathbb{F}_q\).
Repetition code \([n,1,n]\) binary linear code encoding one bit of information into an \(n\)-bit string. Majority decoding requires \(n\) to be odd in order to avoid ties. The idea is to increase the code distance by repeating the logical information several times. It is a \((n,1)\)-Hamming code. Its automorphism group is \(S_n\).
Roth-Lempel code Member of a \(q\)-ary linear code family that includes many examples of MDS codes that are not GRS codes.
Row-Diagonal Parity (RDP) code An MDS array code protecting against double erasures.
Star code An MDS array code protecting against triple erasures.
X-code An MDS array code with a simple geometrical construction that achieves optimal encoding and update complexity.
Ye-Barg code A member of an explicit family of MDS array codes with the optimal access property. The constructions of Ye and Barg achieve optimal repair bandwidth for single-node repair and include optimal-access variants with nearly optimal sub-packetization [5,6].
Zigzag code An MDS array code correcting two erasures with optimal rebuilding ratio; see Ref. [7] for definitions.
\([10,5,6]_9\) Glynn code The unique trace-Hermitian self-dual \([10,5,6]_9\) code, constructed using a 10-arc in \(PG(4,9)\) that is not a rational curve.
\([2^m-1,m,2^{m-1}]\) simplex code A member of the equidistant code family dual to the \([2^m-1,2^m-m-1,3]\) Hamming family.
\([4,2,3]_3\) Tetracode The \([4,2,3]_3\) ternary self-dual MDS code that has connections to lattices [8]. Its weight enumerator is the Gleason polynomial \(g_4\) [9; Rem. 4.2.6].
\([4,2,3]_4\) RS\(_4\) code A Type II Euclidean self-dual extended RS code that is the smallest quaternary extended QR code [10; pg. 296][11; Sec. 2.4.2].
\([6,3,4]_4\) Hexacode The \([6,3,4]_4\) Hermitian self-dual MDS code that has connections to projective geometry, lattices [8], and conformal field theory [12]. Its weight enumerator is the Gleason polynomial \(g_7\) [9; Rem. 4.2.6].
\([7,3,4]\) simplex code Second-smallest nontrivial member of the simplex-code family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(2,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((8,9)\) simplex spherical code under the antipodal mapping. As a simplex code, it is equidistant: every nonzero codeword has Hamming weight \(4\).
\([n,n-1,2]\) Single parity-check (SPC) code An \([n,n-1,2]\) linear binary code whose codewords consist of the message string appended with a parity-check bit or parity bit such that the parity (i.e., sum over all coordinates of each codeword) is zero. If the Hamming weight of a message is odd (even), then the parity bit is one (zero). This code requires only one extra bit of overhead and is therefore inexpensive. Its codewords are all even-weight binary strings, and its parity-check matrix is a row vector of all ones. Its automorphism group is \(S_n\).
\([n,n-1,2]_q\) \(q\)-ary parity-check code An \([n,n-1,2]_q\) linear \(q\)-ary code whose codewords consist of the message string appended with a parity-check or zero-sum check digit such that the sum over all coordinates of each codeword is zero.
\(q\)-ary repetition code An \([n,1,n]_q\) code consisting of codewords \((j,j,\cdots,j)\) for \(j \in \mathbb{F}_q\).
\(q\)-ary simplex code An \([n,m,q^{m-1}]_q\) equidistant projective code with \(n=\frac{q^m-1}{q-1}\), denoted as \(S(q,m)\). The columns of the generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(m-1,q)\), with each column being a chosen representative of the corresponding element. All nonzero simplex codewords have a constant weight of \(q^{m-1}\) [13,14].

References

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[5]
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[6]
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[7]
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[8]
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[13]
A. E.F. Jr. and H. F. Mattson, “Error-correcting codes: An axiomatic approach”, Information and Control 6, 315 (1963) DOI
[14]
E. Weiss, “Linear Codes of Constant Weight”, SIAM Journal on Applied Mathematics 14, 106 (1966) DOI