Completely regular codes in graphs covered by a Hamming graph
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909389977288704 |
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| author | Goryainov, Sergey Krotov, Denis |
| author_facet | Goryainov, Sergey Krotov, Denis |
| contents | In Cayley graphs on the additive group of a small vector space over GF$(q)$, $q=2,3$, we look for completely regular (CR) codes whose parameters are new in Hamming graphs over the same field. The existence of a CR code in such Cayley graph $G$ implies the existence of a CR code with the same parameters in the corresponding Hamming graph that covers $G$. In such a way, we find several completely regular codes with new parameters in Hamming graphs over GF$(3)$. The most interesting findings are two new CR-$1$ (with covering radius~$1$) codes that are independent sets (such CR are equivalent to optimal orthogonal arrays attaining the Bierbrauer--Friedman bound) and one new CR-$2$. By recursive constructions, every knew CR code induces an infinite sequence of CR codes (in particular, optimal orthogonal arrays if the original code was CR-$1$ and independent). In between, we classify feasible parameters of CR codes in several strongly regular graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09698 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Completely regular codes in graphs covered by a Hamming graph Goryainov, Sergey Krotov, Denis Combinatorics In Cayley graphs on the additive group of a small vector space over GF$(q)$, $q=2,3$, we look for completely regular (CR) codes whose parameters are new in Hamming graphs over the same field. The existence of a CR code in such Cayley graph $G$ implies the existence of a CR code with the same parameters in the corresponding Hamming graph that covers $G$. In such a way, we find several completely regular codes with new parameters in Hamming graphs over GF$(3)$. The most interesting findings are two new CR-$1$ (with covering radius~$1$) codes that are independent sets (such CR are equivalent to optimal orthogonal arrays attaining the Bierbrauer--Friedman bound) and one new CR-$2$. By recursive constructions, every knew CR code induces an infinite sequence of CR codes (in particular, optimal orthogonal arrays if the original code was CR-$1$ and independent). In between, we classify feasible parameters of CR codes in several strongly regular graphs. |
| title | Completely regular codes in graphs covered by a Hamming graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.09698 |