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On the construction of $$n$$-dimensional designs from 2-dimensional designs. (English) Zbl 0758.05034
Summary: Let $$H$$ be an abelian group of order $$v$$. If $$X=(f(h_ 1+h_ 2))$$ $$(h_ 1,h_ 2\in H)$$ is a $$v\times v$$ design, then $$X=(f(h_ 1+h_ 2+\cdots+h_ n))$$ is a proper $$n$$-dimensional design. A difficulty with this construction is that it can only be applied to a small number of (2- dimensional) designs. This paper develops a very general technique for generating a proper $$n$$-dimensional design from 2-dimensional designs. Indeed, it is shown that Drake’s generalised Hadamard matrices, Berman’s nega-cyclic and $$\omega$$-cyclic (generalised) weighing matrices and both of the orthogonal designs of order 4 and type (1,1,1,1) can be extended to give proper $$n$$-dimensional designs. In addition, this technique leads to a representation of 2-dimensional designs which generalises the concept of a difference set. This representation is interesting because of its brevity and its wide applicability.

##### MSC:
 05B30 Other designs, configurations 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)