In this paper we study a family of scattered F q -linear sets of rank tn of the projective space P G(2n − 1, q t ) (n ≥ 1, t ≥ 3), called of pseudoregulus type, generalizing results contained in [18] and in [25]. As an application, we characterize, in terms of the associated linear sets, some classical families of semifields: the Generalized Twisted Fields and the 2-dimensional Knuth semifields.

The equivalence problem of F q -linear sets of rank n of PG(1, q n ) is investigated, also in terms of the associated variety, projecting configurations, F q -linear blocking sets of Rédei type and MRD-codes.

a b s t r a c tIn this paper linear sets of finite projective spaces are studied and the ''dual'' of a linear set is introduced. Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids of orthogonal polar spaces and finite semifields. Besides ''old'' results, new ones are proven and some open questions are discussed.

We introduce a family of linear sets of PG(1, q 2n ) arising from maximum scattered linear sets of pseudoregulus type of PG(3, q n ). For n = 3, 4 and for certain values of the parameters we show that these linear sets of PG(1, q 2n ) are maximum scattered and they yield new MRD-codes with parameters (6, 6, q; 5) for q > 2 and with parameters (8, 8, q; 7) for q odd.

Explicit constructions of infinite families of scattered F q -linear sets in P G(r − 1, q t ) of maximal rank rt 2 , for t even, are provided. When q = 2 and r is odd, these linear sets correspond to complete caps in AG(r, 2 t ) fixed by a translation group of size 2 rt 2 . The doubling construction applied to such caps gives complete caps in AG(r + 1, 2 t ) of size 2 rt 2 +1 . For Galois spaces of even dimension greater than 2 and even square order, this solves the long-standing problem of establishing whether the theoretical lower bound for the size of a complete cap is substantially sharp.

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.

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