Scattered trinomials of Fq6[X] in even characteristic

Abstract

In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [9], [24], the authors proved that the trinomial $f_c\left(X\right)=X^q+X^{q^3}+c{X}^{q^5}$ of $F_{q^6}\left[X\right]$ is scattered under the assumptions that q is odd and $c^2+c=1$. They also explicitly observed that this is false when q is even. In this paper, we provide a different set of conditions on c for which this trinomial is scattered in the case of even q. Using tools of algebraic geometry in positive characteristic, we show that when q is even and sufficiently large, there are roughly $q^3$ elements $c\in F_{q^6}$ such that $f_c\left(X\right)$ is scattered. Also, we prove that the corresponding MRD-codes and $F_q$-linear sets of $\mathrm{PG}(1,q^6)$ are not equivalent to the previously known ones.