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 $n\leq 8$ by characterizing three new classes of permutation binomials. In particular one of them has relatively large index $\frac{q^2+q+1}{3}$ over $\mathbb{F}_{q^3}$.

翻译：具有少数项（尤其是置换二项式）的置换多项式因其简单的代数结构而吸引了许多研究者。尽管对置换二项式的研究兴趣浓厚，但其完整刻画仍属未知。本文通过刻画三类新的置换二项式，给出了在 $\mathbb{F}_{2^n}$ 上（其中 $n\leq 8$）形如 $x^i+ax$ 的置换二项式的分类。特别地，其中一类在 $\mathbb{F}_{q^3}$ 上具有相对较大的指数 $\frac{q^2+q+1}{3}$。