Further Results on the Classification of MDS Codes

TL;DR

This paper completes the classification of 7-ary and 8-ary MDS codes with minimum distance at least 3 by leveraging previous classifications and employing computer search techniques.

Contribution

It extends the classification of MDS codes to include all 7-ary and 8-ary cases with specified parameters, filling gaps in existing knowledge.

Findings

Complete classification of 7-ary MDS codes with d≥3.

Complete classification of 8-ary MDS codes with d≥3.

Utilized previous classifications and computer search methods.

Abstract

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.