Abstract
This work is mainly devoted to the study of generalization of Hadamard matrices, first over the sth complex roots of unity and later over any finite group.Relations among them, theorems about existence, and constructions of such matrices are studied.Furthermore, connections with tri-weight extended-BCH codes, strongly regular graphs, relative difference sets, maximal length linear recurring sequences, affine resolvable and symmetrical balanced incomplete block designs, group divisible designs, orthogonal arrays of strength two, affine resolvable partial planes, nets, transversal designs, uniform Klingenberg structures, partial (lamda)-geometries, difference matrices, and positive definite integral Hermitian forms, are also considered.