New lower bounds on the size of (n,r)-arcs in PG(2,q)

TL;DR

This paper improves lower bounds on the maximum size of (n,r)-arcs in projective planes over finite fields by explicit constructions using integer linear programming, some of which meet optimal bounds.

Contribution

It introduces new explicit constructions of (n,r)-arcs in PG(2,q) that improve known lower bounds, utilizing integer linear programming techniques.

Findings

New lower bounds for m_r(2,q) established

Constructed (n,r)-arcs meet the Griesmer bound in some cases

Results obtained through integer linear programming

Abstract

An (n,r)-arc in PG(2,q) is a set of n points such that each line contains at most r of the selected points. It is well-known that (n,r)-arcs in PG(2,q) correspond to projective linear codes. Let m_r(2,q) denote the maximal number n of points for which an (n,r)-arc in PG(2,q) exists. In this paper we obtain improved lower bounds on m_r(2,q) by explicitly constructing (n,r)-arcs. Some of the constructed (n,r)-arcs correspond to linear codes meeting the Griesmer bound. All results are obtained by integer linear programming.