{
  "id": "1702.07939",
  "version": "v1",
  "published": "2017-02-25T19:26:41.000Z",
  "updated": "2017-02-25T19:26:41.000Z",
  "title": "Upper bounds on the smallest size of a saturating set in projective planes and spaces of even dimension",
  "authors": [
    "Daniele Bartoli",
    "Alexander Davydov",
    "Massimo Giulietti",
    "Stefano Marcugini",
    "Fernanda Pambianco"
  ],
  "comment": "14 pages, 34 references, 1 figure",
  "categories": [
    "math.CO",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "In a projective plane $\\Pi_{q}$ (not necessarily Desarguesian) of order $q$, a point subset $\\mathcal{S}$ is saturating (or dense) if any point of $\\Pi_{q}\\setminus \\mathcal{S}$ is collinear with two points in $\\mathcal{S}$. Modifying an approach of [31], we proved the following upper bound on the smallest size $s(2,q)$ of a saturating set in $\\Pi_{q}$: \\begin{equation*} s(2,q)\\leq \\sqrt{(q+1)\\left(3\\ln q+\\ln\\ln q +\\ln\\frac{3}{4}\\right)}+\\sqrt{\\frac{q}{3\\ln q}}+3. \\end{equation*} The bound holds for all q, not necessarily large. By using inductive constructions, upper bounds on the smallest size of a saturating set in the projective space $\\mathrm{PG}(N,q)$ with even dimension $N$ are obtained. All the results are also stated in terms of linear covering codes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-25T19:26:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51E21",
      "51E22",
      "94B05"
    ],
    "keywords": [
      "upper bound",
      "saturating set",
      "smallest size",
      "projective plane",
      "point subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}