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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.)
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