Structure of group invariant weighing matrices of small weight

We show that every weighing matrix of weight n invariant under a finite abelian group G can be generated from a subgroup H of G with |H| ≤ 2n−1. Furthermore, if n is an odd prime power and a proper circulant weighing matrix of weight n and order v exists, then v ≤ 2n−1. We also obtain a lower bound on the weight of group invariant matrices depending on the invariant factors of the underlying