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

Loading next page...
Pleiades Publishing
Copyright © 2014 by Pleiades Publishing, Inc.
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
Publisher site
See Article on Publisher Site


You’re reading a free preview. Subscribe to read the entire article.

DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches


Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.



billed annually
Start Free Trial

14-day Free Trial