A construction for generalized Hadamard matrices

We prove that if pv and pr -1 are both prime powers then there is a generalized Hadamard matrix of order pr(pr -1) with elements from the elementary abelian group Zp x...x Zp. This result was motivated by results of Rajkundlia on BIBD's. This result is then used to produce pr -1 mutually orthogonal F-squares F(pr(pr -1); pr -1).