#### Vol. 2, No. 1, 2005

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

### Giovanna Bonoli and Olga Polverino

Vol. 2 (2005), No. 1, 35–56
DOI: 10.2140/iig.2005.2.35
##### 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 $PG\left(2,{q}^{4}\right)$.

##### Keywords
blocking set, canonical subgeometry, linear set
##### Mathematical Subject Classification 2000
Primary: 05B25, 51E21