New maximum scattered linear sets of the projective line

Abstract

In [2] and [19] are presented the first two families of maximum scattered ${\mathbb{F}}_q$-linear sets of the projective line $\mathrm{PG}(1,q^n)$. More recently in [23] and in [5], new examples of maximum scattered ${\mathbb{F}}_q$-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 ${\mathbb{F}}_q$-linear sets presented in [23] and in [5], for $n=6,8$, are new. Also, for $q$ odd, $q\equiv \pm 1,0(\mathrm{mod}5)$, we present new examples of maximum scattered ${\mathbb{F}}_q$-linear sets in $\mathrm{PG}(1,q^6)$, arising from trinomial polynomials, which define new ${\mathbb{F}}_q$-linear MRD-codes of ${\mathbb{F}}_q^{6\times 6}$ with dimension $12$, minimum distance 5 and middle nucleus (or left idealiser) isomorphic to ${\mathbb{F}}_{q^6}$.