On maximum distance separable and completely regular codes
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| Format: | Preprint |
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2025
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| author | Borges, Joaquim Rifà, Josep Zinoviev, Victor |
| author_facet | Borges, Joaquim Rifà, Josep Zinoviev, Victor |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24292 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On maximum distance separable and completely regular codes Borges, Joaquim Rifà, Josep Zinoviev, Victor Combinatorics Information Theory 94B60, 94B25, 94B05 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. |
| title | On maximum distance separable and completely regular codes |
| topic | Combinatorics Information Theory 94B60, 94B25, 94B05 |
| url | https://arxiv.org/abs/2512.24292 |