Journal of Geometry

, Volume 107, Issue 1, pp 89–117 | Cite as

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

  • Daniele Bartoli
  • Alexander A. Davydov
  • Giorgio Faina
  • Alexey A. Kreshchuk
  • Stefano Marcugini
  • Fernanda Pambianco
Article
First Online:
Received:

Abstract

In the projective plane PG(2, q), upper bounds on the smallest size t 2(2, q) of a complete arc are considered. For a wide region of values of q, the results of computer search obtained and collected in the previous works of the authors and in the present paper are investigated. For q ≤  301813, the search is complete in the sense that all prime powers are considered. This proves new upper bounds on t 2(2, q) valid in this region, in particular
$$\begin{array}{ll}t_{2}(2, q) \;\; < 0.998 \sqrt{3q {\rm ln}\,q} \quad\;\;\,\,{\rm for} \;\; \qquad \quad \;\;7 \leq q \leq 160001;\\ t_{2}(2, q) \;\; < 1.05 \sqrt{3q {\rm ln}\, q}\qquad\,\,{\rm for}\;\; \qquad \quad \;\;7 \leq q \leq 301813;\\ t_{2}(2,q)\;\; < \sqrt{q}{\rm ln}^{0.7295}\,q \qquad \,\,\,\,\,{\rm for} \;\; \quad \quad \,\,\,109 \leq q \leq 160001;\\ t_{2}(2,q) \;\; < \sqrt{q}{\rm ln}^{0.7404}\,q \qquad \,\,\,\,\,{\rm for }\;\;\, \quad 160001 < q \leq 301813.\end{array}$$
The new upper bounds are obtained by finding new small complete arcs in PG(2,q) with the help of a computer search using randomized greedy algorithms and algorithms with fixed (lexicographical) order of points (FOP). Also, a number of sporadic q’s with q ≤  430007 is considered. Our investigations and results allow to conjecture that the 2-nd and 3-rd bounds above hold for all q ≥  109. Finally, random complete arcs in PG(2, q), q ≤  46337, q prime, are considered. The random complete arcs and complete arcs obtained by the algorithm FOP have the same region of sizes; this says on the common nature of these arcs.

Mathematics Subject Classification

Primary 51E21 51E22 Secondary 94B05 

Keywords

Projective planes complete arcs small complete arcs 
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Copyright information

© Springer Basel AG 2015

Authors and Affiliations

  • Daniele Bartoli
    • 1
  • Alexander A. Davydov
    • 3
  • Giorgio Faina
    • 2
  • Alexey A. Kreshchuk
    • 3
  • Stefano Marcugini
    • 2
  • Fernanda Pambianco
    • 2
  1. 1.Department of MathematicsGhent UniversityGentBelgium
  2. 2.Dipartimento di Matematica e InformaticaUniversitá degli Studi di PerugiaPerugiaItaly
  3. 3.Institute for Information Transmission Problems, (Kharkevichinstitute) Russian Academy of SciencesMoscowRussian Federation

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