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A weak difference set construction for higher dimensional designs. (English) Zbl 0838.05019
Authors’ abstract: We continue the analysis of de Launey’s modification of development of designs modulo a finite group $$H$$ by the action of an abelian extension function (AEF), and of the proper higher dimensional designs which result. We extend the characterization of all AEFs from the cyclic group case to the case where $$H$$ is an arbitrary finite abelian group. We prove that our $$n$$-dimensional designs have the form $$(f(j_1j_2\dots j_n))$$, $$j_i\in J$$, where $$J$$ is a subset of cardinality $$|H|$$ of an extension group $$E$$ of $$H$$. We say these designs have a weak difference set construction. We show that two well-known constructions for orthogonal designs fit this development scheme and hence exhibit families of such Hadamard matrices, weighing matrices and orthogonal designs of order $$v$$ for which $$|E|= 2v$$. In particular, we construct proper higher dimensional Hadamard matrices for all orders $$4t\leq 100$$, and conference matrices of order $$q+ 1$$ where $$q$$ is an odd prime power. We conjecture that such Hadamard matrices exist for all orders $$v\equiv 0\bmod 4$$.

MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) 05B99 Designs and configurations
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