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

Abstract

A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the $q$-polynomial over ${\mathbb{F}}_{q^6}$, $q\equiv 1(\mathrm{mod}4)$ described in arXiv:1906.05611, arXiv:1910.02278 is generalized for any even $n\ge 6$ to an ${\mathbb{F}}_q$-linear automorphism $\psi (x)$ of ${\mathbb{F}}_q^n$ of order $n$. Such $\psi (x)$ and some functional powers of it are proved to be scattered. In particular this provides new maximum scattered linear sets of the projective line $\mathrm{PG}(1,q^n)$ for $n=8,10$. The polynomials described in this paper lead to a new infinite family of MRD-codes in ${\mathbb{F}}_q^{n\times n}$ with minimum distance $n-1$ for any odd $q$ if $n\equiv 0(\mathrm{mod}4)$ and any $q\equiv 1(\mathrm{mod}4)$ if $n\equiv 2(\mathrm{mod}4)$.