Groups with maximal irredundant covers and minimal blocking sets

TL;DR

This paper establishes a new construction method for minimal blocking sets in higher-dimensional projective spaces by leveraging properties of groups with maximal irredundant covers, linking combinatorial and algebraic structures.

Contribution

It proves a novel result connecting minimal blocking sets in different projective spaces through group-theoretic methods, expanding understanding of their structure.

Findings

Constructs minimal blocking sets in higher dimensions from lower-dimensional ones.

Links properties of blocking sets to group theory with maximal irredundant covers.

Provides a new algebraic approach to combinatorial geometry problems.

Abstract

Let $n$ be a positive integer. Denote by $\mathrm{PG}(n,q)$ the $n$-dimensional projective space over the finite field $\mathbb{F}_q$ of order $q$. A blocking set in $\mathrm{PG}(n,q)$ is a set of points that has non-empty intersection with every hyperplane of $\mathrm{PG}(n,q)$. A blocking set is called minimal if none of its proper subsets are blocking sets. In this note we prove that if $\mathrm{PG}(n_i,q)$ contains a minimal blocking set of size $k_i$ for $i\in\{1,2\}$, then $\mathrm{PG}(n_1+n_2+1,q)$ contains a minimal blocking set of size $k_1+k_2-1$. This result is proved by a result on groups with maximal irredundant covers.