Cocyclic Hadamard matrices and difference sets

Abstract 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.

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