Further results on permutation pentanomials over Fq3 in characteristic two

Let $q=2^m$. In a recent paper [34], Zhang and Zheng investigated 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 $F_{q^3}(d_0=1,2,4)$ with a certain linearized polynomial $L\left(x\right)$. They applied the multivariate method and specific techniques to analyze the number of solutions of certain equations, and proposed an open problem: the permutation property of some pentanomials of this form remains unproven. In this paper, inspired by the idea of [12], we further characterize the permutation property of such pentanomials over $F_{q^3}(d_0=1,2,4)$. The techniques presented in this paper will be useful for investigating more new classes of permutation polynomials.