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Upper bounds on the smallest size of a complete arc in PG(2, q) under a certain probabilistic conjecture

Problems of Information Transmission volume 50, pages 320–339 (2014)Cite this article

Abstract

In the projective plane PG(2, q), we consider an iterative construction of complete arcs which adds a new point in each step. It is proved that uncovered points are uniformly distributed over the plane. For more than half of steps of the iterative process, we prove an estimate for the number of newly covered points in every step. A natural (and well-founded) conjecture is made that the estimate holds for the other steps too. As a result, we obtain upper bounds on the smallest size t 2(2, q) of a complete arc in PG(2, q), in particular,

$$\begin{array}{*{20}c} {t_2 (2,q) < \sqrt q \sqrt {3\ln q + \ln \ln q + \ln 3} + \sqrt {\frac{q} {{3\ln q}}} + 3,} \\ {t_2 (2,q) < 1.87\sqrt {q\ln q} .} \\ \end{array}$$

Nonstandard types of upper bounds on t 2(2, q) are considered, one of them being new. The effectiveness of the new bounds is illustrated by comparing them with the smallest known sizes of complete arcs obtained in recent works of the authors and in the present paper via computer search in a wide region of q. We note a connection of the considered problems with the so-called birthday problem (or birthday paradox).

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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, Università degli Studi di Perugia, Perugia, Italy

    D. Bartoli, G. Faina, S. Marcugini & F. Pambianco

  2. Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

    D. Bartoli, A. A. Davydov & A. A. Kreshchuk

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  1. D. Bartoli
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  2. A. A. Davydov
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  3. G. Faina
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  4. A. A. Kreshchuk
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  5. S. Marcugini
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  6. F. Pambianco
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Original Russian Text © D. Bartoli, A.A. Davydov, G. Faina, A.A. Kreshchuk, S. Marcugini, F. Pambianco, 2014, published in Problemy Peredachi Informatsii, 2014, Vol. 50, No. 4, pp. 22–42.

Supported in part by the Ministry for Education, University and Research of Italy (MIUR), project “Geometrie di Galois e strutture di incidenza,” and Italian National Group for Algebraic and Geometric Structures and their Applications (G.N.S.A.G.A.).

Supported by the European Community under a Marie-Curie Intra-European Fellowship, FACE project no. 626511.

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Bartoli, D., Davydov, A.A., Faina, G. et al. Upper bounds on the smallest size of a complete arc in PG(2, q) under a certain probabilistic conjecture. Probl Inf Transm 50, 320–339 (2014). https://doi.org/10.1134/S0032946014040036

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