A new lower bound for the size of an affine blocking set

TL;DR

This paper improves the lower bound on the size of blocking sets in affine planes of order at least 25, establishing that such sets must contain at least q + floor(sqrt(q)) + 3 points, advancing prior results from the 1980s.

Contribution

The paper presents a new, tighter lower bound for the size of blocking sets in affine planes of order q ≥ 25, surpassing previous bounds from the 1980s.

Findings

Blocking sets in affine planes of order q ≥ 25 have at least q + floor(sqrt(q)) + 3 points.

The new bound improves upon the classical lower bounds established in the 1980s.

The result applies to non-desarguesian affine planes, broadening its relevance.

Abstract

A blocking set in an affine plane is a set of points $B$ such that every line contains at least one point of $B$. The best known lower bound for blocking sets in arbitrary (non-desarguesian) affine planes was derived in the 1980's by Bruen and Silverman. In this note, we improve on this result by showing that a blocking set of an affine plane of order $q$, $q\geq 25$, contains at least $q+\lfloor\sqrt{q}\rfloor+3$ points.