On the non-existence of some generalized Hadamard matrices

A conjecture for generalized Hadamard matrices over group G of order p states that Hadamard matrix GH(p, h) exists only if the matrices In and nIn are Hermitian congruent [1], where n = ph and p is prime. References (4,5] document many parameter values for which non-existence is known to occur. Here, methods for establishing non-existence based upon a fundamental necessary condition of Brock [2] are considered. Several parameter sequences for which non-existence occurs are identified. The methods exploited complement de Launey's [6] approach via number theoretic properties of the Hadamard determinant. Neither investigation is exhaustive of all possibilities.