More classes of permutation pentanomials over finite fields with even characteristic

Abstract

Let $q=2^m$. In this paper, we propose several classes of permutation pentanomials of the form ${ϵ}_0{x}^{d_0}+L({ϵ}_1{x}^{d_1}+{ϵ}_2{x}^{d_2})$ over $F_{q^n}(2\le n\le 4)$ from some certain linearized polynomial $L\left(x\right)$ by using multivariate method and some techniques to determine the solutions of some equations. Furthermore, two classes of permutation pentanomials over $F_{q^n}$ for n satisfying $3|q^n-1$ are also constructed based on some bijections over the unit circle $U_{\tau }$ of $F_{q^n}$ with order $\tau =q^{n-1}+q^{n-2}+\cdots +q^2+q+1$.