Several new permutation quadrinomials over finite fields of odd characteristic

Abstract

Let $\mathbb {F}_{q}$ denote the finite field with q elements. In this paper, we study the permutation property of the polynomial $x^3+ax^{q+2}+bx^{2q+1}+cx^{3q} \in \mathbb {F}_{q^{2}}[x]$, where char ${{\mathbb {F}}}_{q}=3,5$. More precisely, for $a,b,c\in {{\mathbb {F}}}_{q}^*$, we propose eight new classes of permutation quadrinomials of the form $x^3+ax^{q+2}+bx^{2q+1}+cx^{3q} \in \mathbb {F}_{q^{2}}[x]$ for $q=3^m$, and three new classes of permutation quadrinomials of the form $x^3+ax^{q+2}+bx^{2q+1}+cx^{3q} \in \mathbb {F}_{q^{2}}[x]$ for $q=5^m$.