{
  "id": "1312.2155",
  "version": "v3",
  "published": "2013-12-08T00:04:08.000Z",
  "updated": "2015-06-28T10:37:03.000Z",
  "title": "Tables, bounds and graphics of the smallest known sizes of complete arcs in the plane $\\mathrm{PG}(2,q)$ for all $q\\le160001$ and sporadic $q$ in the interval $[160801\\ldots 430007]$",
  "authors": [
    "Daniele Bartoli",
    "Alexander A. Davydov",
    "Giorgio Faina",
    "Alexey A. Kreshchuk",
    "Stefano Marcugini",
    "Fernanda Pambianco"
  ],
  "comment": "85 pages, 11 figures, 6 tables, 83 references; data and figures are updated, the region of data is increased; the title is changed; figures and references are added",
  "categories": [
    "math.CO"
  ],
  "abstract": "In the projective planes $\\mathrm{PG}(2,q)$, we collect the smallest known sizes of complete arcs for the regions \\begin{align*} &\\mbox{all } q\\le160001,~~ q \\mbox{ prime power};\\\\ &Q_{4}=\\{34 \\mbox{ sporadic }q'\\mbox{s in the interval }[160801\\ldots430007], \\mbox{ see Table 3}\\}. \\end{align*} For $q\\le160001$, the collection of arc sizes is complete in the sense that arcs for all prime powers are considered. This proves new upper bounds on the smallest size $t_{2}(2,q)$ of a complete arc in $\\mathrm{PG}(2,q)$, in particular \\begin{align*} t_{2}(2,q)&<0.998\\sqrt{3q\\ln q}<1.729\\sqrt{q\\ln q}&\\mbox{ for }&&7&\\le q\\le160001;~~(1) \\\\ t_{2}(2,q)&<\\sqrt{q}\\ln^{0.7295}q&\\mbox{ for }&&109&\\le q\\le160001;~~(2)\\\\ t_{2}(2,q)&<\\sqrt{q}\\ln^{c_{up}(q)}q,~~c_{up}(q)=\\frac{0.27}{\\ln q}+0.7,&\\mbox{ for }&&19&\\le q\\le160001;~~(3)\\\\ t_{2}(2,q)&<0.6\\sqrt{q}\\ln^{\\varphi_{up}(q;0.6)} q,~~\\varphi_{up}(q;0.6)=\\frac{1.5}{\\ln q}+0.802,&\\mbox{ for }&&19&\\le q\\le160001.~~(4) \\end{align*} Moreover, the bounds (2) -- (4) hold also for $q\\in Q_{4}$. Also, \\begin{align*} t_{2}(2,q)&<1.006\\sqrt{3q\\ln q}<1.743\\sqrt{q\\ln q}&\\mbox{ for }&&q\\in Q_{4}.~~(5) \\end{align*} Our investigations and results allow to conjecture that the bounds (2) -- (5) hold for all $q\\geq109$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-05T16:32:16.000Z",
      "title": "Tables of sizes of small complete arcs in the plane $PG(2,q)$, $q\\le 410009$",
      "abstract": "In the projective planes $PG(2,q)$, we collect the smallest known sizes $\\overline{t}_{2}(2,q)$ of complete arcs for the region \\begin{align} &T=\\{2\\le q\\le49727, q \\mbox{ power prime}\\}\\,\\cup \\{2\\le q\\leq 150001, q \\mbox{ prime}\\}\\,\\,\\cup\\notag\\\\ &\\{40 \\mbox{ sporadic prime }q's \\mbox{ in the interval }[150503\\ldots410009], \\mbox{ see Table 6}\\}.\\notag \\end{align} Basing on the collected sizes we obtain the following upper bounds on the smallest size $t_{2}(2,q)$ of a complete arc in $PG(2,q)$. $$t_{2}(2,q)\\le1.745\\sqrt{q\\ln q},~q\\in T.$$ $$t_{2}(2,q)\\le\\sqrt{q}\\ln ^{0.7295}q,~q\\ge109,~q\\in T.$$ $$t_{2}(2,q)\\le\\sqrt{q}\\ln^{f(q)}q,\\quad f(q)=\\frac{0.27}{\\ln q}+0.7,\\quad q\\in T.$$ $$t_{2}(2,q)\\le0.6\\sqrt{q}\\ln^{\\varphi(q)}q,\\quad \\varphi(q)=\\frac{1.5}{\\ln q}+0.802,\\quad q\\in T.$$ Our researches and results allow us to conjecture that these bounds hold for all $q\\ge109$.",
      "comment": "89 pages, 6 figures, 6 tables, 73 references; data and figures are updated, the region of data is increased; in the title, 360007 is changed by 410009; figures and references are added",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-06-28T10:37:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51E21",
      "51E22",
      "94B05"
    ],
    "keywords": [
      "small complete arcs",
      "upper bounds",
      "sporadic prime",
      "bounds hold",
      "collected sizes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 85,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.2155B"
    }
  }
}