Towards a classification of permutation binomials of the form \(x^i+ax\) over \({\mathbb {F}}_{2^n}\)

Permutation polynomials with few terms (especially permutation binomials) attract many people due to their simple algebraic structure. Despite the great interests in the study of permutation binomials, a complete characterization of permutation binomials is still unknown. Let \(q=2^n\) $q=2^n$ for a positive integer n. In this paper, we start classifying permutation binomials of the form \(x^i+ax\) $x^i+ax$ over \({\mathbb {F}}_{q}\) ${\mathbb{F}}_q$ in terms of their indices. After carrying out an exhaustive search of these permutation binomials over \({\mathbb {F}}_{2^n}\) ${\mathbb{F}}_{2^n}$ for n up to 12, we gave three new infinite classes of permutation binomials over \({\mathbb {F}}_{q^2}\) ${\mathbb{F}}_{q^2}$ , \({\mathbb {F}}_{q^3}\) ${\mathbb{F}}_{q^3}$ , and \({\mathbb {F}}_{q^4}\) ${\mathbb{F}}_{q^4}$ respectively, for \(q=2^n\) $q=2^n$ with arbitrary positive integer n. In particular, these binomials over \({\mathbb {F}}_{q^3}\) ${\mathbb{F}}_{q^3}$ have relatively large index \(\frac{q^2+q+1}{3}\) $\frac{q^2+q+1}{3}$ . As an application, we can completely explain all the permutation binomials of the form \(x^i+ax\) $x^i+ax$ over \({\mathbb {F}}_{2^n}\) ${\mathbb{F}}_{2^n}$ for \(n\le 8\) $n\le 8$ . Moreover, we prove that there does not exist permutation binomials of the form \(x^{2q^3+2q^2+2q+3}+ax\) $x^{2q^3+2q^2+2q+3}+ax$ over \({\mathbb {F}}_{q^4}\) ${\mathbb{F}}_{q^4}$ such that \(a\in {\mathbb {F}}_{q^4}^*\) $a\in {\mathbb{F}}_{q^4}^{\ast }$ and \(n=2\,m\) $n=2m$ with \(m\ge 2\) $m\ge 2$ .