A q-ary code of length n, size M, and minimum distanced is called an (n,M,d)q code. An (n,qk,n - k + 1)qcode 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, qk, d)qcode 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)5code and the (n, 7n-2, 3)7MDS 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, 8n-2, 3)8codes for n = 6, 7, 8, and 9, respectively. One of the equivalence classes of perfect (9, 87, 3)8codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction.