Let q = 2 m . In this paper, we propose several classes of permutation pentanomials of the form ϵ 0 x d 0 + L ( ϵ 1 x d 1 + ϵ 2 x d 2 ) over F q n ( 2 ≤ n ≤ 4 ) from some certain linearized polynomial L ( x ) by using multivariate method and some techniques to determine the solutions of some equations. Furthermore, two classes of permutation pentanomials over F q n for n satisfying 3 | q n − 1 are also constructed based on some bijections over the unit circle U τ of F q n with order τ = q n − 1 + q n − 2 + ⋯ + q 2 + q + 1 .
Let q = 2...
