The rank of a scattered $\mathbb{ F}_q$-linear set of $\mathrm{ PG}(r-1,q^n)$, $rn$ even, is at most $rn/2$ as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of $r$, $n$, $q$ ($rn$ even) for scattered $\mathbb{ F}_q$-linear sets of rank $rn/2$. In this paper we prove that the bound $rn/2$ is sharp also in the remaining open cases. Recently Sheekey proved that scattered $\mathbb{ F}_q$-linear sets of $\mathrm{ PG}(1,q^n)$ of maximum rank $n$ yield $\mathbb{ F}_q$-linear MRD-codes with dimension $2n$ and minimum distance $n-1$. We generalize this result and show that scattered $\mathbb{ F}_q$-linear sets of $\mathrm{ PG}(r-1,q^n)$ of maximum rank $rn/2$ yield $\mathbb{ F}_q$-linear MRD-codes with dimension $rn$ and minimum distance $n-1$.
Categories: math.CO