{
  "id": "0907.0002",
  "version": "v1",
  "published": "2009-07-01T19:59:11.000Z",
  "updated": "2009-07-01T19:59:11.000Z",
  "title": "On the binary codes with parameters of doubly-shortened 1-perfect codes",
  "authors": [
    "Denis Krotov"
  ],
  "comment": "12pp",
  "journal": "Des. Codes Cryptogr. 57(2) 2010, 181-194",
  "doi": "10.1007/s10623-009-9360-5",
  "categories": [
    "math.CO",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "We show that any binary $(n=2^m-3, 2^{n-m}, 3)$ code $C_1$ is a part of an equitable partition (perfect coloring) $\\{C_1,C_2,C_3,C_4\\}$ of the $n$-cube with the parameters $((0,1,n-1,0)(1,0,n-1,0)(1,1,n-4,2)(0,0,n-1,1))$. Now the possibility to lengthen the code $C_1$ to a 1-perfect code of length $n+2$ is equivalent to the possibility to split the part $C_4$ into two distance-3 codes or, equivalently, to the biparticity of the graph of distances 1 and 2 of $C_4$. In any case, $C_1$ is uniquely embeddable in a twofold 1-perfect code of length $n+2$ with some structural restrictions, where by a twofold 1-perfect code we mean that any vertex of the space is within radius 1 from exactly two codewords.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-01T19:59:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "94B25"
    ],
    "keywords": [
      "binary codes",
      "parameters",
      "possibility",
      "structural restrictions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.0002K"
    }
  }
}