  In this paper, we construct a new class of complete permutation monomials and
several classes of permutation polynomials. Further, by giving another
characterization of o-polynomials, we obtain a class of permutation polynomials
of the form $G(x)+ \gamma Tr(H(x))$, where G(X) is neither a permutation nor a
linearized polynomial. This is an answer to the open problem 1 of Charpin and
Kyureghyan in [P. Charpin and G. Kyureghyan, When does $G(x)+ \gamma Tr(H(x))$
permute $\mathbb{F}_{p^n}$?, Finite Fields and Their Applications 15 (2009)
615--632].
