On cocyclic weighing matrices and the regular group actions of certain paley matrices

In this paper we consider cocyclic weighing matrices. Cocyclic development of a weighing matrix is shown to be related to regular group actions on the points of the associated group divisible design. We show that a cocyclic w eighing matrix is equivalent to a relative difference set with central forb idden subgroup of order two. We then set out an agenda for studying a known cocyclic weighing matrix and carry it out for the Paley conference matrix and for the type I Paley Hadamard matrix. Using a connection with certain n ear fields, we determine all the regular group actions on the group divisib le design associated to such a Paley matrix. It happens that all the regula r actions of the Paley type I Hadamard matrix have already been described i n the literature, however, new regular actions are identified for the Paley conference matrix. This allows us to determine all the extension groups an d indexing groups for the cocycles of the aforementioned Paley matrices, an d gives new families of normal and non-normal relative difference sets with forbidden subgroup of size two. (C) 2000 Elsevier Science B.V. All rights reserved.