Linear sets and MRD-codes arising from a class of scattered linearized polynomials

A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the q-polynomial over F_(q^6), q≡1(mod4) described in Bartoli et al. (ARS Math Contemp 19:125–145, 2020) and Zanella and Zullo (Discrete Math 343:111800, 2020) is generalized for any even n≥ 6 to an Fq-linear automorphism ψ(x) of Fqn of order n. Such ψ(x) and some functional powers of it are proved to be scattered. In particular, this provides new maximum scattered linear sets of the projective line PG(1,qn) for n= 8 , 10. The polynomials described in this paper lead to a new infinite family of MRD-codes in Fqn×n with minimum distance n- 1 for any odd q if n≡0(mod4) and any q≡1(mod4) if n≡2(mod4).