{
  "id": "1410.7184",
  "version": "v1",
  "published": "2014-10-27T11:10:12.000Z",
  "updated": "2014-10-27T11:10:12.000Z",
  "title": "Symmetric bilinear forms over finite fields with applications to coding theory",
  "authors": [
    "Kai-Uwe Schmidt"
  ],
  "comment": "33 pages",
  "categories": [
    "math.CO",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\\mathbb{F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a commutative translation association scheme. We give explicit expressions for the eigenvalues of this scheme in terms of linear combinations of generalised Krawtchouk polynomials. We then study $d$-codes in this scheme, namely subsets $Y$ of $X(m,q)$ with the property that, for all distinct $A,B\\in Y$, the rank of $A-B$ is at least $d$. We prove bounds on the size of a $d$-code and show that, under certain conditions, the inner distribution of a $d$-code is determined by its parameters. Constructions of $d$-codes are given, which are optimal among the $d$-codes that are subgroups of $X(m,q)$. Finally, with every subset $Y$ of $X(m,q)$, we associate two classical codes over $\\mathbb{F}_q$ and show that their Hamming distance enumerators can be expressed in terms of the inner distribution of $Y$. As an example, we obtain the distance enumerators of certain cyclic codes, for which many special cases have been previously obtained using long ad hoc calculations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-27T11:10:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A63",
      "05E30",
      "11T71",
      "94B15"
    ],
    "keywords": [
      "symmetric bilinear forms",
      "finite fields",
      "coding theory",
      "applications",
      "long ad hoc calculations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.7184S"
    }
  }
}