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Cocyclic generalised Hadamard matrices and central relative difference sets. (English) Zbl 0919.05007
The main result is the equivalence of the existence of: (1) a $$G$$-cocyclic generalized Hadamard matrix GH$$(w,v/w)$$ with entries in $$C$$, (2) a relative $$(v,w,v,v/w)$$-difference set in a central extension $$E$$ of $$C$$ by $$G$$ relative to $$C$$, and (3) a square divisible $$(v,w,v,v/w)$$-design, class regular with respect to $$C$$ with a central extension $$E$$ of $$C$$ by $$G$$ as a regular group of automorphisms, where $$G$$ is a finite group of order $$v$$, and $$C$$ is a finite abelian group of order $$w$$ such that $$w| v$$.
This nice theorem generalizes a result of Jungnickel on splitting relative $$(v,w,v,v/w)$$-difference sets and a result of Launey on $$G$$-cocyclic Hadamard matrices. In the last section the authors look at constructions and restrictions on the parameters for $$G$$-cocyclic generalized Hadamard matrices.

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