New results on linear permutation polynomials with coefficients in a subfield

Abstract

Some families of linear permutation polynomials of ${\mathbb{F}}_{q^{ms}}$ with coefficients in ${\mathbb{F}}_{q^m}$ are explicitly described (via conditions on their coefficients) as isomorphic images of classical subgroups of the general linear group of degree $m$ over the ring $\frac{{\mathbb{F}}_q[x]}{⟨x^s-1⟩}$. In addition, the sizes of some of these families are computed. Finally, several criteria to construct linear permutation polynomials of ${\mathbb{F}}_{q^{2p}}$ (where $p$ is a prime number) with prescribed coefficients in ${\mathbb{F}}_{q^2}$ are given. Examples illustrating the main results are presented.