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

Upper bounds on the smallest size of a complete arc in PG(2, q) under a certain probabilistic... 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

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

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Publisher
Springer Journals
Copyright
Copyright © 2014 by Pleiades Publishing, Inc.
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946014040036
Publisher site
See Article on Publisher Site

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).

Journal

Problems of Information TransmissionSpringer Journals

Published: Jan 7, 2015

References

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