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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.

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