New sizes of complete arcs in PG(2,q)

TL;DR

This paper presents new upper bounds on the smallest size of complete arcs in projective planes PG(2,q) for various q, achieved through computational search with randomized greedy algorithms, improving known results.

Contribution

The paper introduces new upper bounds and small complete arcs in PG(2,q) for specific q values using computational methods, advancing the understanding of arc sizes.

Findings

New upper bounds for t_{2}(2,q) for 853<= q<= 2879 and some larger q.

Verification that t_{2}(2,q)<4.5√q for certain q values.

Presentation of new small complete arcs obtained via computer search.

Abstract

New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853<= q<= 2879 and q=3511,4096, 4523,5003,5347,5641,5843,6011. For q<= 2377 and q=2401,2417,2437, the relation t_{2}(2,q)<4.5\sqrt{q} holds. The bounds are obtained by finding of new small complete arcs with the help of computer search using randomized greedy algorithms. Also new sizes of complete arcs are presented.