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

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Mathematical Subject Classification 2000

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