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.

In [2] and [19] are presented the first two families of maximum scattered F q -linear sets of the projective line PG(1, q n ). More recently in [23] and in [5], new examples of maximum scattered F q -subspaces of V (2, q n ) have been constructed, but the equivalence problem of the corresponding linear sets is left open.Here we show that the F q -linear sets presented in [23] and in [5], for n = 6, 8, are new. Also, for q odd, q ≡ ±1, 0 (mod 5), we present new examples of maximum scattered F q -linear sets in PG(1, q 6 ), arising from trinomial polynomials, which define new F q -linear MRD-codes of F 6×6 q with dimension 12, minimum distance 5 and middle nucleus (or left idealiser) isomorphic to F q 6 .

We provide sufficient and necessary conditions for the coefficients of a q-polynomial f over F q n which ensure that the number of distinct roots of f in F q n equals the degree of f . We say that these polynomials have maximum kernel. As an application we study in detail q-polynomials of degree q n−2 over F q n which have maximum kernel and for n ≤ 6 we list all q-polynomials with maximum kernel. We also obtain information on the splitting field of an arbitrary qpolynomial. Analogous results are proved for q s -polynomials as well, where gcd(s, n) = 1.

In this paper we investigate the geometric properties of the configuration consisting of a k-subspace Γ and a canonical subgeometry Σ in PG(n − 1, q n ), with Γ ∩ Σ = ∅. The idea motivating is that such properties are reflected in the algebraic structure of the linear set which is projection of Σ from the vertex Γ. In particular we deal with the maximum scattered linear sets of the line PG(1, q n ) found by Lunardon and Polverino in [17] and recently generalized by Sheekey in [21]. Our aim is to characterize this family by means of the properties of the vertex of the projection as done by Csajbók and the first author of this paper for linear sets of pseudoregulus type. With reference to such properties, we construct new examples of scattered linear sets in PG(1, q 6 ), yielding also to new examples of MRD-codes in F 6×6 q with left idealiser isomorphic to F q 6 .1 Angle brackets without the indication of a field will denote the projective span of a set of points in a projective space.

Let V be an r-dimensional F q n -vector space. We call an F q -subspace U of V h-scattered if U meets the h-dimensional F q n -subspaces of V in F q -subspaces of dimension at most h. In 2000 Blokhuis and Lavrauw proved that dim Fq U ≤ rn/2 when U is 1-scattered. Subspaces attaining this bound have been investigated intensively because of their relations with projective two-weight codes and strongly regular graphs. MRD-codes with a maximum idealiser have also been linked to rn/2-dimensional 1-scattered subspaces and to n-dimensional (r − 1)scattered subspaces.In this paper we prove the upper bound rn/(h+1) for the dimension of h-scattered subspaces, h > 1, and construct examples with this dimension. We study their intersection numbers with hyperplanes, introduce a duality relation among them, and study the equivalence problem of the corresponding linear sets.

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For any admissible value of the parameters n and k there exist [n, k]-Maximum Rank Distance F q -linear codes. Indeed, it can be shown that if field extensions large enough are considered, almost all rank metric codes are MRD. On the other hand, very few families up to equivalence of such codes are currently known. In the present paper we study some invariants of MRD codes and evaluate their value for the known families, providing a new characterization of generalized twisted Gabidulin codes.