Cocyclic Development of Designs

We present the basic theory of cocyclic development of designs, in which group development over a finite group G is modified by the action of a cocycle defined on G x G. Negacyclic and w-cyclic development are both special cases of cocyclic development. Techniques of design construction using the group ring, arising from difference set methods, also apply to cocyclic designs. Important classes of Hadamard matrices and generalized weighing matrices are cocyclic. We derive a characterization of cocyclic development which allows us to generate all matrices which are cocyclic over G. Any cocyclic matrix is equivalent to one obtained by entrywise action of an asymmetric matrix and a symmetric matrix on a G-developed matrix. The symmetric matrix is a Kronecker product of back w-cyclic matrices, and the asymmetric matrix is determined by the second integral homology group of G. We believe this link between combinatorial design theory and low-dimensional group cohomology leads to (i) a new way to generate combinatorial designs; (ii) a better understanding of the structure of some known designs; and (iii) a better understanding of known construction techniques.