{
  "id": "1404.0469",
  "version": "v2",
  "published": "2014-04-02T06:26:19.000Z",
  "updated": "2015-07-31T09:13:38.000Z",
  "title": "Tables, bounds and graphics of sizes of complete lexiarcs in the plane $\\mathrm{PG}(2,q)$ for all $q\\le301813$ and sporadic $q$ in the interval $[301897\\ldots430007]$ obtained by an algorithm with fixed order of points (FOP)",
  "authors": [
    "Daniele Bartoli",
    "Alexander A. Davydov",
    "Giorgio Faina",
    "Alexey A. Kreshchuk",
    "Stefano Marcugini",
    "Fernanda Pambianco"
  ],
  "comment": "114 pages, 14 figures, 5 tables, 88 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 previous works of the authors, a step-by-step algorithm FOP which uses any $fixed~order~of~points$ in the projective plane $\\mathrm{PG}(2,q)$ is proposed for constructing small complete arcs. In every step, the algorithm adds to a running arc the first point in the fixed order not lying on the bisecants of the arc. In this work we $\\textbf{collect the sizes of complete lexiarcs}$ (i.e. complete arcs in $\\mathrm{PG}(2,q)$ obtained by the algorithm FOP with the lexicographical orders of points) in the following regions: $\\textbf{all}$ $q\\le301813$, $q$ prime power; 23 sporadic $q$'s in the interval $[301897\\ldots430007]$, see Table 2. In the work [9], the smallest $known$ sizes of complete arcs in $\\mathrm{PG}(2,q)$ are collected for all $q\\leq160001$, $q$ prime power. The sizes of complete arcs collected in this work and in the work [9] provide the following upper bounds on the smallest size $t_{2}(2,q)$ of a complete arc in the projective plane $\\mathrm{PG}(2,q)$: \\begin{align*} t_{2}(2,q)&< 0.998\\sqrt{3q\\ln q}<1.729\\sqrt{q\\ln q}&\\mbox{ for }&&7&\\le q\\le160001; \\\\ t_{2}(2,q)&<1.05\\sqrt{3q\\ln q}<1.819\\sqrt{q\\ln q}&\\mbox{ for }&&7&\\le q\\le301813;\\\\ t_{2}(2,q)&<\\sqrt{q}\\ln^{0.7295}q&\\mbox{ for }&&109&\\le q\\le160001;\\\\ t_{2}(2,q)&<\\sqrt{q}\\ln^{0.7404}q&\\mbox{ for }&&160001&<q\\le301813. \\end{align*} Our investigations and results allow to conjecture that the 2-nd and 3-rd bounds above hold for all $q\\geq109$. In addition, it is shown that sizes of complete lexiarcs in $\\mathrm{PG}(2,q)$, $q\\gtrapprox90000$, exceed the smallest known sizes of complete arcs by only at most $6\\%$. It is noted also that sizes of the random complete arcs and complete lexiarcs behave similarly. This work can be considered as a continuation and development of the paper [10].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-02T06:26:19.000Z",
      "title": "Tables of sizes of small complete arcs in the plane $PG(2,q)$, $q\\le 190027$, obtained by an algorithm with fixed order of points (FOP)",
      "abstract": "In the recent works of the authors, an algorithm FOP using any fixed order of points in $PG(2,q)$ is proposed for constructing small complete arcs. The algorithm is based on an intuitive postulate that $PG(2,q)$ contains a sufficient number of relatively small complete arcs. Also, in these works, it is shown that the type of order on the points of $PG(2,q)$ is not relevant. In this work we collect the sizes of complete arcs obtained by the algorithm FOP with the lexicographical and the Singer orders of points in the following regions: Lexicographical order: $3\\le q\\le67993$, $q$ prime; Lexicographical order: 43 sporadic prime $q$'s in the interval $[69997\\ldots190027]$. Singer order: $5\\le q\\le40009$, $q$ prime.",
      "comment": "66 pages, 2 figures, 3 tables, 58 references; Tables are attached also as .csv files. To view attachments, please download and extract the gzipped tar source file listed under \"Other formats\"",
      "journal": null,
      "doi": null,
      "authors": [
        "Daniele Bartoli",
        "Alexander A. Davydov",
        "Giorgio Faina",
        "Stefano Marcugini",
        "Fernanda Pambianco"
      ]
    },
    {
      "version": "v2",
      "updated": "2015-07-31T09:13:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51E21",
      "51E22",
      "94B05"
    ],
    "keywords": [
      "fixed order",
      "singer order",
      "algorithm fop",
      "relatively small complete arcs",
      "constructing small complete arcs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 114,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.0469B"
    }
  }
}