{
  "id": "1511.03623",
  "version": "v1",
  "published": "2015-11-11T19:44:06.000Z",
  "updated": "2015-11-11T19:44:06.000Z",
  "title": "Inclusion Matrices and the MDS Conjecture",
  "authors": [
    "Simeon Ball",
    "Ameera Chowdhury"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let F_q be a finite field of order q with characteristic p. An arc is an ordered family of vectors in (F_q)^k in which every subfamily of size k is a basis of (F_q)^k. The MDS conjecture, which was posed by Segre in 1955, states that if k <= q, then an arc in (F_q)^k has size at most q+1, unless q is even and k=3 or k=q-1, in which case it has size at most q+2. We propose a conjecture which would imply that the MDS conjecture is true for almost all values of k when q is odd. We prove our conjecture in two cases and thus give simpler proofs of the MDS conjecture when k <= p, and if q is not prime, for k <= 2p-2. To accomplish this, given an arc G of (F_q)^k and a nonnegative integer n, we construct a matrix M_G^{\\uparrow n}, which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix M_G^{\\uparrow n} to properties of the arc G and may provide new tools in the computational classification of large arcs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-11T19:44:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A03",
      "05B35",
      "51E21",
      "94B05"
    ],
    "keywords": [
      "mds conjecture",
      "inclusion matrix",
      "main results relate algebraic properties",
      "finite field",
      "large arcs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151103623B"
    }
  }
}