Maximum scattered linear sets and MRD-codes

The rank of a scattered $F_q$-linear set of $PG(r-1,q^n)$, $rn$ 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 $F_q$-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 $F_q$-linear sets of $PG(1,q^n)$ of maximum rank $n$ yield $F_q$--linear MRD-codes with dimension $2n$ and minimum distance $n-1$. We generalize this result and show that scattered $F_q$-linear sets of $PG(r-1,q^n)$ of maximum rank $rn/2$ yield $F_q$--linear MRD-codes with dimension $rn$ and minimum distance $n-1$.