Further Results on the Classification of MDS Codes

A $q$-ary maximum distance separable (MDS) code $C$ with length $n$, dimension $k$ over an alphabet $\mathcal{A}$ of size $q$ is a set of $q^k$ codewords that are elements of $\mathcal{A}^n$, such that the Hamming distance between two distinct codewords in $C$ is at least $n-k+1$. Sets of mutually orthogonal Latin squares of orders $q\leq 9$, corresponding to two-dimensional \mbox{$q$-}ary MDS codes, and $q$-ary one-error-correcting MDS codes for $q\leq 8$ have been classified in earlier studies. These results are used here to complete the classification of all $7$-ary and $8$-ary MDS codes with $d\geq 3$ using a computer search.