Journal of Algebra 291 (2005) 551-606. doi:10.1016/j.jalgebra.2005.02.005Received by publisher: 2004-09-14Harvest Date: 2016-01-04 12:19:59DOI: 10.1016/j.jalgebra.2005.02.005Page Range: 551-60

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.

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Abstract-An explicit construction of a family of binary LDPC codes called LU(3, q), where q is a power of a prime, was recently given. A conjecture was made for the dimensions of these codes when q is odd. The conjecture is proved in this note. The proof involves the geometry of a 4-dimensional symplectic vector space and the action of the symplectic group and its subgroups.