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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Ghent University, Krijgslaan 281-B, Gent, 9000, Belgium

    Daniele Bartoli

  2. Dipartimento di Matematica e Informatica, Universitá degli Studi di Perugia, Via Vanvitelli 1, Perugia, 06123, Italy

    Giorgio Faina, Stefano Marcugini & Fernanda Pambianco

  3. Institute for Information Transmission Problems, (Kharkevichinstitute) Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, GSP-4, Moscow, 127994, Russian Federation

    Alexander A. Davydov & Alexey A. Kreshchuk

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  1. Daniele Bartoli
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  2. Alexander A. Davydov
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  3. Giorgio Faina
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  4. Alexey A. Kreshchuk
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  5. Stefano Marcugini
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Correspondence to Alexander A. Davydov.

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The research of D. Bartoli, G. Faina, S. Marcugini, and F. Pambianco was supported in part by Ministry for Education, University and Research of Italy (MIUR) (Project “Geometrie di Galois e strutture di incidenza”) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INDAM). The work of D. Bartoli was also supported by the European Community under a Marie-Curie Intra-European Fellowship (FACE Project: number 626511). The research of A.A. Davydov and A.A. Kreshchuk was made in IITP RAS and supported by a Grant from the Russian Science Foundation (Project No. 14-50-00150).

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Bartoli, D., Davydov, A.A., Faina, G. et al. Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search. J. Geom. 107, 89–117 (2016). https://doi.org/10.1007/s00022-015-0277-z

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