A classification of permutation binomials of the form $x^i+ax$ over ${\mathbb{F}}_{2^n}$ for dimensions up to 8

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. In this paper, we give a classification of permutation binomials of the form $x^i+ax$ over ${\mathbb{F}}_{2^n}$, where $n\le 8$ by characterizing three new classes of permutation binomials. In particular one of them has relatively large index $\frac{q^2+q+1}{3}$ over ${\mathbb{F}}_{q^3}$.