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Journal of Geometry

, Volume 106, Issue 1, pp 1–17 | Cite as

New types of estimates for the smallest size of complete arcs in a finite Desarguesian projective plane

  • Daniele BartoliEmail author
  • Alexander A. Davydov
  • Giorgio Faina
  • Stefano Marcugini
  • Fernanda Pambianco
Article

Abstract

New types of upper bounds for the smallest size t 2(2, q) of a complete arc in the projective plane PG(2, q) are proposed. The value \({t_{2}(2, q) = d(q)\sqrt{q} \ln q}\), where d(q) <  1 is a decreasing function of q, is used. The case \({d(q) < \alpha/ \ln{\beta q} + \gamma}\), where \({\alpha,\beta,\gamma}\) are positive constants independent of q, is considered. It is shown that
$$t_{2}(2, q) < (2/\,{\rm ln} \frac{1}{10}q + 0.32)\sqrt{q}\, {\rm ln}\, q\, {\rm if} \, q \leq 67993, q \,{\rm prime}, {\rm and }\, q \in R,$$
where R is a set of 27 values in the region 69997...110017. Also, for \({q \in [9311,67993]}\), q prime, and \({q \in R}\), it is shown that
$$\sqrt{q}({\rm ln}\, q)^{a_1-bq} < t_{2}(2, q) < \sqrt{q}({\rm ln}\, q)^{a_2-bq},$$
\({a_1=0.771, a_2=0.752, b=2.2 \cdot 10^{-7}.}\) In addition, our results allow us to conjecture that these estimates hold for all q. An algorithm FOP using any fixed order of points in PG(2, q) is proposed for constructing complete arcs. The algorithm is based on an intuitive postulate that PG(2, q) contains a sufficient number of relatively small complete arcs. It is shown that the type of order on the points of PG(2, q) is not relevant.

Mathematics Subject Classification (2010)

Primary 51E21 51E22 Secondary 94B05 

Keywords

Projective planes complete arcs smallcomplete arcs upper bounds 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Daniele Bartoli
    • 1
    Email author
  • Alexander A. Davydov
    • 2
  • Giorgio Faina
    • 1
  • Stefano Marcugini
    • 1
  • Fernanda Pambianco
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly
  2. 2.Institute for Information Transmission Problems (Kharkevich Institute)Russian Academy of SciencesMoscowRussian Federation

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