On subspaces defining linear sets of maximum rank

Abstract

Let $V$ denote an $r$-dimensional ${\mathbb{F}}_{q^n}$-vector space. Let $U$ and $W$ be ${\mathbb{F}}_q$-subspaces of $V$, $L_U$ and $L_W$ the projective points of $\mathrm{PG}(V,q^n)$ defined by $U$ and $W$ respectively. We address the problem when $L_W=L_U$ under the hypothesis that $U$ and $W$ have maximum dimension, i.e., ${\dim }_{{\mathbb{F}}_q}W={\dim }_{{\mathbb{F}}_q}U=$ $rn-n$, and we give a complete characterization for $r=2$.