A modified power spectral density test applied to weighing matrices with small weight

The power spectral density test has been used for at least a decade in the search for many kinds of combinatorial matrices, such as weighing matrices for instance. In this paper we establish a modified power spectral density test that we apply to the search for weighing matrices of small weights constructed from two circulants. The main novelty of our approach is to define the Discrete Fourier Transform on the support of the first rows of the two circulants, thus exploiting the inherent sparsity of the problem. This new formalism turns out to be very efficient for small weights 9,18,36 and we find 10 new weighing matrices W(2a <...p,18) for prime pa{37,47,53,59,61,67,73,79,83,97}. These matrices are given here for the first time. We also discuss briefly a connection with Combinatorial Optimization.