$\mathbb{F}_q$-linear blocking sets in $\mathrm{PG}(2,q^4)$

Abstract

An ${\mathbb{F}}_q$-linear blocking set $B$ of $\pi =PG\left(2,q^n\right)$, $q=p^h$, $n>2$, can be obtained as the projection of a canonical subgeometry $\mathrm{\Sigma }\simeq PG\left(n,q\right)$ of ${\mathrm{\Sigma }}^{\ast }=PG\left(n,q^n\right)$ to $\pi$ from an $\left(n-3\right)$-dimensional subspace $\mathrm{\Lambda }$ of ${\mathrm{\Sigma }}^{\ast }$, disjoint from $\mathrm{\Sigma }$, and in this case we write $B=B_{\mathrm{\Lambda },\mathrm{\Sigma }}$. In this paper we prove that two ${\mathbb{F}}_q$-linear blocking sets, $B_{\mathrm{\Lambda },\mathrm{\Sigma }}$ and $B_{{\mathrm{\Lambda }}^{\prime },{\mathrm{\Sigma }}^{\prime }}$, of exponent $h$ are isomorphic if and only if there exists a collineation $\phi$ of ${\mathrm{\Sigma }}^{\ast }$ mapping $\mathrm{\Lambda }$ to ${\mathrm{\Lambda }}^{\prime }$ and $\mathrm{\Sigma }$ to ${\mathrm{\Sigma }}^{\prime }$. This result allows us to obtain a classification theorem for ${\mathbb{F}}_q$-linear blocking sets of the plane $PG\left(2,q^4\right)$.