Groups with maximal irredundant covers and minimal blocking sets

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