Some results on weighing matrices

It is shown that if q is a prime power then there exists a circulant weighing matrix of order q2 + q + 1 with q2 non-zero elements per row and column. This result allows the bound N to be lowered in the theorem of Geramita and Wallis that " given a square integer k there exists an integer N dependent on k such that weighing matrices of weight k and order n and orthogonal designs (1, k) of order 2n exist for every n > N".