Cyclic Difference Sets And Cyclic Hadamard Matrices

The collection of cyclic Hadamard matrices {H = (a_{i - j}) : 0 <= i, j < n, and a_i = -1, 1} of order n is characterized by the orthogonality relation HH^T = nI. Only two of such matrices are currently known. It will be shown that this collection consists of precisely two matrices. An application of this result implies that there are exactly seven Barker sequences over the binary set {-1, 1}.