On maximum distance separable and completely regular codes
Joaquim Borges, Josep Rifà, Victor Zinoviev
TL;DR
This work tackles the problem of identifying when an $MDS$ code over $\mathbb{F}_q$ is completely regular ($CR$) or uniformly packed in the wide sense ($UPWS$). By organizing the analysis around code length regimes ($n=q+2$, $n=q+1$, and $n\le q\le 5$) and leveraging bounds such as the Griesmer and MDS constraints, the authors obtain complete classifications for the extremal lengths and a full small-field taxonomy, including precise results on self-dual CR codes with $\rho\le 3$. They reveal infinite families of CR $MDS$ codes (e.g., certain Hamming and Reed–Solomon extensions) and enumerate sporadic small-field examples, clarifying when $CR$ coincides with $UPWS$. The findings contribute to the understanding of the intersection of optimal distance properties and regularity in linear codes, with implications for code construction and theory. All results are presented with detailed field- and parameter-dependent distinctions, offering a comprehensive map of when $MDS$ codes can be completely regular across different $q$ and $n$.
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 $ρ\leq 3$: 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.