Maximum scattered linear sets and MRD-codes - Csajbók, Bence; Marino, Giuseppe; Polverino, Olga; Zullo, Ferdinando (Journal of algebraic combinatorics, 2017)
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 2 n 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 .