{
  "id": "1211.3248",
  "version": "v3",
  "published": "2012-11-14T09:31:09.000Z",
  "updated": "2013-05-05T02:35:17.000Z",
  "title": "Lower bounds on maximal determinants of +-1 matrices via the probabilistic method",
  "authors": [
    "Richard P. Brent",
    "Judy-anne H. Osborn",
    "Warren D. Smith"
  ],
  "comment": "32 pages, 64 references, 1 table. Theorem 4 added in v2. Minor improvements/corrections in v3",
  "categories": [
    "math.CO"
  ],
  "abstract": "We show that the maximal determinant D(n) for $n \\times n$ ${\\pm 1}$-matrices satisfies $R(n) := D(n)/n^{n/2} \\ge \\kappa_d > 0$. Here $n^{n/2}$ is the Hadamard upper bound, and $\\kappa_d$ depends only on $d := n-h$, where $h$ is the maximal order of a Hadamard matrix with $h \\le n$. Previous lower bounds on R(n) depend on both $d$ and $n$. Our bounds are improvements, for all sufficiently large $n$, if $d > 1$. We give various lower bounds on R(n) that depend only on $d$. For example, $R(n) \\ge 0.07 (0.352)^d > 3^{-(d+3)}$. For any fixed $d \\ge 0$ we have $R(n) \\ge (2/(\\pi e))^{d/2}$ for all sufficiently large $n$ (and conjecturally for all positive $n$). If the Hadamard conjecture is true, then $d \\le 3$ and $\\kappa_d \\ge (2/(\\pi e))^{d/2} > 1/9$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-05-05T02:35:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B20",
      "05D40",
      "15B34"
    ],
    "keywords": [
      "lower bounds",
      "maximal determinant",
      "probabilistic method",
      "sufficiently large",
      "hadamard upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.3248B"
    }
  }
}