Groups With Maximal Irredundant Covers And Minimal Blocking Sets

Abstract

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