Transgression and the calculation of cocyclic matrices

It is conjectured that binary cocyclic matrices are a uniform source of Hadamard matrices. In testing this conjecture, it is useful to have a general method of calculating cocyclic matrices. We present such a method in this paper. The method draws on standard cohomology theory of finite groups. In particular we employ the Universal Coefficient Theorem, which expresses the second cohomology group explicitly as an internal direct sum of two subgroups. One subgroup arises the image of a transgn~ssllon homomorphism. The method reduces essentially to determination of (representative cocycles for) the image of transgression. There is a resultant description of a given cocyclic matrix as the Hadamard product of certain matrices. The factors in the product generally are not canonically determined, and this may be significant in the development of algorithms for calculating co cyclic Hadamard matrices. An example is given to illustrate the method.