On permutation binomials

Abstract

Let $\mathbb{F}_{q}$ be the finite field of characteristic $p$ containing $q = p^{r}$ elements. Let $f(x)=ax^{n} + x^{m}$ be a binomial with coefficients in $\mathbb{F}_{q}$ and $d = \mbox{gcd\,}(n-m,q-1)$. In this paper, we prove that there does not exist any permutation binomial such that $d$ satisfies certain congruence conditions, and we do some computations to list all non permutation binomials for $n-m=3$ and $q\leq 100$.