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, 5
⁴
, 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, 8
⁷
, 3)
₈
codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.