Some results about permutation properties of a kind of binomials over finite fields

Abstract

Permutation polynomials have many applications in finite fields theory, coding theory, cryptography, combinatorial design, communication theory, and so on. Permutation binomials of the form $x^{r}(x^{q-1}+a)$ over $\mathbb{F}_{q^2}$ have been studied before, K. Li, L. Qu and X. Chen proved that they are permutation polynomials if and only if $r=1$ and $a^{q+1}\not=1$. In this paper, we consider the same binomial, but over finite fields $\mathbb{F}_{q^3}$ and $\mathbb{F}_{q^e}$. Two different kinds of methods are employed, and some partial results are obtained for them.