{
  "id": "2312.08360",
  "version": "v1",
  "published": "2023-12-13T18:52:05.000Z",
  "updated": "2023-12-13T18:52:05.000Z",
  "title": "On the existence of some completely regular codes in Hamming graphs",
  "authors": [
    "Denis S. Krotov"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "We solve several first questions in the table of small parameters of completely regular (CR) codes in Hamming graphs $H(n,q)$. The most uplifting result is the existence of a $\\{13,6,1;1,6,9\\}$-CR code in $H(n,2)$, $n\\ge 13$. We also establish the non-existence of a $\\{11,4;3,6\\}$-code and a $\\{10,3;4,7\\}$-code in $H(12,2)$ and $H(13,2)$. A partition of the complement of the quaternary Hamming code of length~$5$ into $4$-cliques is found, which can be used to construct completely regular codes with covering radius $1$ by known constructions. Additionally we discuss the parameters $\\{24,21,10;1,4,12\\}$ of a putative completely regular code in $H(24,2)$ and show the nonexistence of such a code in $H(8,4)$. Keywords: Hamming graph, equitable partition, completely regular code",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-13T18:52:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "94B25"
    ],
    "keywords": [
      "regular code",
      "hamming graph",
      "small parameters",
      "quaternary hamming code",
      "cr code"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}