Classification of MDS Codes over Small Alphabets

Abstract

A q-ary code of length n, size M, and minimum distance d is called an (n, M, d) _q code. An $(n,q^{k},d)_{q}$ code with $d = n - k + 1$ is said to be maximum distance separable (MDS). Here we show that every code with parameters $(k + d - 1,q^{k},d)_{q}$ where k, d ≥ 3 and q = 5, 7, is equivalent to a linear code, which implies that the $(6,5^{4},3)_{5}$ code and the $(n,7^{n-2},3)_{7}$ codes for n = 6, 7, 8 are unique. We also show that there are 14, 8, 4, and 4 equivalence classes of $(n,8^{n-2},3)_{8}$ codes for n = 6, 7, 8, 9, respectively. This work is continuation of a previous article classifying $(5,q^{3},3)_{q}$ codes for q = 5, 7, 8.