Blocking Planes by Lines in $\operatorname{PG}(n,q)$

TL;DR

This paper investigates the minimal number of lines in finite projective spaces needed to cover all planes, providing improved upper bounds through a recursive construction scheme.

Contribution

It introduces a refined recursive construction method that improves upper bounds for the minimal blocking sets in $ ext{PG}(n,q)$, advancing understanding of $(2,1)$-blocking sets.

Findings

Improved upper bounds for the size of blocking sets.

A recursive construction scheme for $(2,1)$-blocking sets.

Enhanced bounds applicable to general cases.

Abstract

In this paper, we study the cardinality of the smallest set of lines of the finite projective spaces $\operatorname{PG}(n,q)$ such that every plane is incident with at least one line of the set. This is the first main open problem concerning the minimum size of $(s,t)$-blocking sets in $\operatorname{PG}(n,q)$, where we set $s=2$ and $t=1$. In $\operatorname{PG}(n,q)$, an $(s,t)$-blocking set refers to a set of $t$-spaces such that each $s$-space is incident with at least one chosen $t$-space. This is a notoriously difficult problem, as it is equivalent to determining the size of certain $q$-Tur\'an designs and $q$-covering designs. We present an improvement on the upper bounds of Etzion and of Metsch via a refined scheme for a recursive construction, which in fact enables improvement in the general case as well.