Hadamard Matrices from Relative Difference Sets

is the transpose of H and I is the identity matrix. It is easily shown that for H to exist n must be 1, 2, or a multiple of 4 [2]. The converse problem of constructing Hadamard matrices of all possible orders is much more difficult. Many authors have made contributions in an effort to find a solution, the results being many and varied (a list of all the constructions known in 1972 is contained in [5]). It is the purpose of this paper to add yet another class of Hadamard matrices to the ever growing list. More precisely, we prove the following results.