Cocyclic Hadamard matrices and difference sets

This paper locates cocyclic Hadamard matrices within the mainstream of combinatorial design theory. We prove that the existence of a cocyclic Hadamard matrix of order 4t is equivalent to the existence of a normal relative difference set with parameters (4t,2,4t,2t). In the basic case we note there is a corresponding equivalence between coboundary Hadamard matrices and Menon-Hadamard difference sets. These equivalences unify and explain results in the theories of Hadamard groups, divisible designs with regular automorphism groups, and periodic autocorrelation functions.