Circulant weighing matrices

Abstract

Circulant weighing matrices are matrices with entries in {-1,0,1} where the rows are pairwise orthogonal and each successive row is obtained from the previous row by a fixed cyclic permutation. They are useful in solving problems where it is necessary to determine as accurately as possible, the "weight" of n "objects" in n "weighings". They have also been successfully used to improve the performance of certain optical instruments such as spectrometers and image scanners. In this thesis I discuss the basic properties of circulant weighing matrices, prove most of the known existence results known to me at the time of writing this thesis and classify the circulant weighing matrices with precisely four nonzero entries in each row. The problem of classifying all circulant weighing matrices is related to the "cyclic projective plane problem". This relationship is established and I have devoted the final chapter of this thesis to cyclic projective planes and their relationship to circulant weighing matrices. The final theorem in this thesis yields information about equations of the kind xy¯²=a in cyclic projective planes. Show more