  In \cite{D1}, Dickson listed all permutation polynomials up to degree 5 over
an arbitrary finite field, and all permutation polynomials of degree 6 over
finite fields of odd characteristic. The classification of degree 6 permutation
polynomials over finite fields of characteristic 2 was left incomplete. In this
paper we complete the classification of permutation polynomials of degree 6
over finite fields of characteristic 2. In addition, all permutation
polynomials of degree 7 over finite fields of characteristic 2 are classified.
