{
  "id": "math/0509164",
  "version": "v1",
  "published": "2005-09-08T11:09:00.000Z",
  "updated": "2005-09-08T11:09:00.000Z",
  "title": "Groebner bases and combinatorics for binary codes",
  "authors": [
    "M. Borges-Quintana",
    "M. A. Borges-Trenard",
    "P. Fitzpatrick",
    "E. Martinez-Moro"
  ],
  "comment": "Submitted to Appl. Algebra Engrg. Comm. Comput",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gr\\\"obner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for the code. By associating the code with the set of cycles in a graph, we can solve the problem of finding all codewords of minimal length (minimal cycles in a graph), and show how to find a minimal cycle basis. Finally we discuss some results on the computation of the Gr\\\"obner basis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-09-08T11:09:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13P10",
      "94B05"
    ],
    "keywords": [
      "binary codes",
      "groebner bases",
      "combinatorics",
      "binary linear code",
      "minimal cycle basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9164B"
    }
  }
}