Maximum scattered linear sets and MRD-codes

The rank of a scattered $${\mathbb F}_q$$Fq-linear set of $${{\mathrm{{PG}}}}(r-1,q^n)$$PG(r-1,qn), 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 $${\mathbb F}_q$$Fq-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 $${\mathbb F}_q$$Fq-linear sets of $${{\mathrm{{PG}}}}(1,q^n)$$PG(1,qn) of maximum rank n yield $${\mathbb F}_q$$Fq-linear MRD-codes with dimension 2n and minimum distance $$n-1$$n-1. We generalize this result and show that scattered $${\mathbb F}_q$$Fq-linear sets of $${{\mathrm{{PG}}}}(r-1,q^n)$$PG(r-1,qn) of maximum rank rn / 2 yield $${\mathbb F}_q$$Fq-linear MRD-codes with dimension rn and minimum distance $$n-1$$n-1.