There is a Paley graph for each prime power q such that q ≡ 1 (mod 4). The vertex set is the field F q , and two vertices x and y are joined by an edge if and only if x − y is a nonzero square of F q . We compute the Smith normal forms of the adjacency matrix and Laplacian matrix of a Paley graph.

Abstract. We determine the Smith normal forms of the incidence matrices of points and projective (r − 1)-dimensional subspaces of PG(n, q) and of the incidence matrices of points and r-dimensional affine subspaces of AG(n, q) for all n, r, and arbitrary prime power q.

In Dickson (1896Dickson ( -1897 [2], the author 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.

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