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

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.

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Bartoli, D., Davydov, A.A., Faina, G. et al. New types of estimates for the smallest size of complete arcs in a finite Desarguesian projective plane. J. Geom. 106, 1–17 (2015). https://doi.org/10.1007/s00022-014-0224-4

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Mathematics Subject Classification (2010)

  • Primary 51E21
  • 51E22
  • Secondary 94B05

Keywords

  • Projective planes
  • complete arcs
  • smallcomplete arcs
  • upper bounds