Saturating linear sets in PG$(2,q^4)$

Abstract

Bonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study \emph{$1$-saturating} linear sets in PG$(2,q^4)$, that is $\mathbb{F}_q$-linear sets in PG$(2,q^4)$ with the property that their secant lines cover the entire plane. By making use of a characterization of generalized Gabidulin codes, we prove that the rank of such a linear set is at least $5$. This answers to a recent question posed by Bartoli, Borello and Marino.