On Circulant Weighing Matrices

Algebraic techniques are employed to obtain necessary conditions for the existence of certain circulant weighing matrices. As an application we rule out the existence of many circulant weighing matrices. We study orders n = s2 + s + 1, for 10 less than or equal to s less than or equal to 25. These orders correspond to the number of points in a projective plane of order s.