an:00068554
Zbl 0758.05034
de Launey, Warwick
On the construction of \(n\)-dimensional designs from 2-dimensional designs
EN
Australas. J. Comb. 1, 67-81 (1990).
00008643
1990
j
05B30 05B20 05B10
\(n\)-dimensional designs; 2-dimensional designs; Hadamard matrices; weighing matrices; orthogonal designs; difference set
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.