Non-splitting Abelian (4 t, 2, 4 t, 2 t) Relative Difference Sets and Hadamard Cocycles

Using cohomology we show that in studying the existence of an abelian non-splitting (4 t, 2, 4 t, 2 t) relative difference set, D, we can assume the groups in question have a certain simple form. We obtain an explicit constructive equivalence between generalized perfect binary arrays and cocycles that define Hadamard matrices and thereby show directly that the existence of D corresponds to that of a symmetric Hadamard matrix of a certain form. This extends the well-known equivalence in the case of splitting relative difference sets.