On the Classification of MDS Codes

A q-ary code of length n, size M, and minimum distanced is called an (n,M,d)q code. An (n,q^k,n - k + 1)_q code is called a maximum distance separable (MDS) code. In this paper, some MDS codes over small alphabets are classified. It is shown that every (k + d - 1, q^k, d)_q code with k ≥ 3, d ≥ 3, q ∈ (5, 7} is equivalent to a linear code with the same parameters. This implies that the (6, 5⁴, 3)₅ code and the (n, 7^n-2, 3)₇ MDS codes for n ∈ (6, 7, 8} are unique. The classification of one-error-correcting 8-ary MDS codes is also finished; there are 14, 8, 4, and 4 equivalence classes of (n, 8^n-2, 3)₈ codes for n = 6, 7, 8, and 9, respectively. One of the equivalence classes of perfect (9, 8⁷, 3)₈ codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.