On maximum distance separable and completely regular codes

TL;DR

This paper classifies when maximum distance separable (MDS) codes over finite fields are also completely regular, providing complete classifications for certain lengths and field sizes, and clarifying the existence of specific self-dual codes.

Contribution

It offers new classifications of MDS codes that are also completely regular or uniformly packed, especially for lengths q+1, q+2, and small q, filling gaps in previous research.

Findings

Complete classification for lengths n=q+1 and n=q+2.

No nontrivial MDS codes for q=2.

Four nontrivial MDS families for q=4.

Abstract

We investigate when a maximum distance separable ($MDS$) code over $F_q$ is also completely regular ($CR$). For lengths $n=q+1$ and $n=q+2$ we provide a complete classification of the $MDS$ codes that are $CR$ or at least uniformly packed in the wide sense ($UPWS$). For the more restricted case $n\leq q$ with $q\leq 5$ we obtain a full classification (up to equivalence) of all nontrivial $MDS$ codes: there are none for $q=2$; only the ternary Hamming code for $q=3$; four nontrivial families for $q=4$; and exactly six linear $MDS$ codes for $q=5$ (three of which are $CR$ and one admits a self-dual version). Additionally, we close two gaps left open in a previous classification of self-dual $CR$ codes with covering radius $\rho\leq 3$: we precisely determine over which finite fields the $MDS$ self-dual completely regular codes with parameters $[2,1,2]_q$ and $[4,2,3]_q$ exist.