Towards a classification of permutation binomials of the form xi + ax over F2n

Permutation polynomials with few terms (especially permutation binomials) attract manypeople due to their simple algebraic structure. Despite the great interests in the study of per-mutation binomials, a complete characterization of permutation binomials is still unknown.Let q=2(n) for a positive integern. In this paper, we start classifying permutation binomialsof the form x(i)+ax over F-q in terms of their indices. After carrying out an exhaustive searchof these permutation binomials over F-2n fornup to 12, we gave three new infinite classes of permutation binomials over F-q(2),F-q(3),and F-q(4) respectively, for q=2(n) with arbitrary positiveintegern. In particular, these binomials over F-q(3) have relatively large index (q2+q+1)(3).Asanapplication, we can completely explain all the permutation binomials of the form x(i)+ax over F-2(n) for n <= 8. Moreover, we prove that there does not exist permutation binomials of the form x(2 q3+2q2+2q+3+)ax over F-q(4) such that a is an element of F-q4* and n=2 m with m >= 2.