Some new results on circulant weighing matrices (Q1610679)

A circulant weighing matrix \(C(n,k)\) is a circulant matrix \(C\) of size \(n\) with entries \(\{0,\pm 1\}\in \mathbb{Z}\) such that \(C\cdot C^T=k\cdot \mathbf{I}_n\) (\(\mathbf{I}_n\) identity matrix). It is interesting to obtain necessary and sufficient conditions on the existence of \(C(n,k)\)'s. Such conditions are known only for \(k=4\) and \(9\). The paper contains a nontrivial (though standard) application of algebraic number theory to obtain some necessary conditions in the cases \(k=25, 49\) and \(121\). Unfortunately, only finitely many examples can be ruled out.