{
  "id": "1905.04978",
  "version": "v1",
  "published": "2019-05-13T11:25:01.000Z",
  "updated": "2019-05-13T11:25:01.000Z",
  "title": "Small weight code words arising from the incidence of points and hyperplanes in PG($n,q$)",
  "authors": [
    "Sam Adriaensen",
    "Lins Denaux",
    "Leo Storme",
    "Zsuzsa Weiner"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $C_{n-1}(n,q)$ be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG($n,q$). Recently, Polverino and Zullo \\cite{polverino} proved that within this code, all non-zero code words of weight at most $2q^{n-1}$ are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We improve this result, proving that when $q>17$ and $q\\notin\\{25,27,29,31,32,49,121\\}$, all code words of weight at most $(4q-\\sqrt{8q}-\\frac{33}{2})q^{n-2}$ are linear combinations of incidence vectors of hyperplanes through a fixed $(n-3)$-space. Depending on the omitted value for $q$, we can lower the bound on the weight of $c$ to obtain the same results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-13T11:25:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B25",
      "94B05"
    ],
    "keywords": [
      "small weight code words arising",
      "hyperplane",
      "incidence vector",
      "desarguesian projective space pg",
      "non-zero code words"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}