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

AbstractIn 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 weighing matrix is equivalent to a relative difference set with central forbidden 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 near fields, we determine all the regular group actions on the group divisible design associated to such a Paley matrix. It happens that all the regular actions of the Paley type I Hadamard matrix have already been described in the literature, however, new regular actions are identified for the Paley conference matrix. This allows us to determine all the extension groups and indexing groups for the cocycles of the aforementioned Paley matrices, and gives new families of normal and non-normal relative difference sets with forbidden subgroup of size two