{
  "id": "2411.09698",
  "version": "v1",
  "published": "2024-11-14T18:59:01.000Z",
  "updated": "2024-11-14T18:59:01.000Z",
  "title": "Completely regular codes in graphs covered by a Hamming graph",
  "authors": [
    "Sergey Goryainov",
    "Denis Krotov"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-14T18:59:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hamming graph",
      "regular codes",
      "optimal orthogonal arrays",
      "knew cr code induces",
      "cayley graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}