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.

翻译：本文研究了有限域 $F_q$ 上最大距离可分（$MDS$）码何时也是完全正则（$CR$）码的问题。对于码长 $n=q+1$ 和 $n=q+2$ 的情形，我们完整分类了那些是 $CR$ 码或至少是广义一致填充（$UPWS$）码的 $MDS$ 码。对于限制更强的情形 $n\leq q$ 且 $q\leq 5$，我们获得了所有非平凡 $MDS$ 码的完整分类（在等价意义下）：当 $q=2$ 时不存在；当 $q=3$ 时仅有三元汉明码；当 $q=4$ 时有四个非平凡码族；当 $q=5$ 时恰好有六个线性 $MDS$ 码（其中三个是 $CR$ 码，一个允许自对偶版本）。此外，我们填补了先前关于覆盖半径 $ρ\leq 3$ 的自对偶 $CR$ 码分类中的两个空白：我们精确确定了参数为 $[2,1,2]_q$ 和 $[4,2,3]_q$ 的 $MDS$ 自对偶完全正则码存在于哪些有限域上。