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 \(q=2^n\) for a positive integer n. In this paper, we start classifying permutation binomials of the form \(x^i+ax\) over \({\mathbb {F}}_{q}\) in terms of their indices. After carrying out an exhaustive search of these permutation binomials over \({\mathbb {F}}_{2^n}\) for n up to 12, we gave three new infinite classes of permutation binomials over \({\mathbb {F}}_{q^2}\), \({\mathbb {F}}_{q^3}\), and \({\mathbb {F}}_{q^4}\) respectively, for \(q=2^n\) with arbitrary positive integer n. In particular, these binomials over \({\mathbb {F}}_{q^3}\) have relatively large index \(\frac{q^2+q+1}{3}\). As an application, we can completely explain all the permutation binomials of the form \(x^i+ax\) over \({\mathbb {F}}_{2^n}\) for \(n\le 8\). Moreover, we prove that there does not exist permutation binomials of the form \(x^{2q^3+2q^2+2q+3}+ax\) over \({\mathbb {F}}_{q^4}\) such that \(a\in {\mathbb {F}}_{q^4}^*\) and \(n=2\,m\) with \(m\ge 2\).