Partial λ-Geometries and Generalized Hadamard Matrices Over Groups

Section 1 of this paper contains all the work which deals exclusively with generalizations of Hadamard matrices. The non-existence theorem proven here (Theorem 1.10) generalizes a theorem of Hall and Paige [15] on the non-existence of complete mappings in certain groups. In Sections 2 and 3, we consider the duals of (Hanani) transversal designs; these dual structures, which we call (s, r, µ)-nets, are a natural generalization of the much studied (Bruck) nets which in turn are equivalent to sets of mutually orthogonal Latin squares. An (s, r, µ)-net is a set ofs2µpoints together withrparallel classes of blocks. Each class consists ofsblocks of equal cardinality. Two non-parallel blocks meet in preciselyµpoints. It has been proven thatris always less than or equal to (s2µ– l) / (s– 1).