{
  "id": "1212.1056",
  "version": "v1",
  "published": "2012-12-05T15:38:10.000Z",
  "updated": "2012-12-05T15:38:10.000Z",
  "title": "Geometric representations of binary codes embeddable in three dimensions",
  "authors": [
    "Pavel Rytíř"
  ],
  "comment": "21 pages, 19 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "We say that a binary linear code C has a geometric representation if there exists a two dimensional simplicial complex D such that C is a punctured code of the kernel ker D of the incidence matrix of D and dim C = dim ker D. We show that every binary linear code has a geometric representation that can be embedded into R^4. Moreover, we show that a binary linear code C has a geometric representation in R^3 if and only if there exists a graph G such that C equals the cut space of G. This is a polynomially testable property and hence we can conclude that there is a polynomial algorithm that decides the minimal dimension of a geometric representation of a binary linear code.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-05T15:38:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "94B05",
      "90C27",
      "55U10"
    ],
    "keywords": [
      "geometric representation",
      "binary linear code",
      "binary codes embeddable",
      "dimensional simplicial complex",
      "minimal dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.1056R"
    }
  }
}