{
  "id": "1512.03361",
  "version": "v1",
  "published": "2015-12-10T18:33:35.000Z",
  "updated": "2015-12-10T18:33:35.000Z",
  "title": "MDS codes in the Doob graphs",
  "authors": [
    "Evgeny Bespalov",
    "Denis Krotov"
  ],
  "comment": "In Russian, 30 pp",
  "categories": [
    "math.CO",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "The Doob graph $D(m,n)$, where $m>0$, is the direct product of $m$ copies of The Shrikhande graph and $n$ copies of the complete graph $K_4$ on $4$ vertices. The Doob graph $D(m,n)$ is a distance-regular graph with the same parameters as the Hamming graph $H(2m+n,4)$. In this paper we consider MDS codes in Doob graphs with code distance $d \\ge 3$. We prove that if $2m+n>6$ and $2<d<2m+n$, then there are no MDS codes with code distance $d$. We characterize all MDS codes with code distance $d \\ge 3$ in Doob graphs $D(m,n)$ when $2m+n \\le 6$. We characterize all MDS codes in $D(m,n)$ with code distance $d=2m+n$ for all values of $m$ and $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-10T18:33:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "doob graph",
      "mds codes",
      "code distance",
      "complete graph",
      "distance-regular graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "ru",
      "license": "arXiv",
      "status": "editable"
    }
  }
}