Higher dimensional orthogonal designs and applications

The concept of orthogonal design is extended to higher dimensions. A proper g -dimensional design [d_{ijk \cdots \upsilon}] is defined as one in which all parallel (g-1) -dimensional layers, in any orientation parallel to a hyper plane, are uncorrelated. This is equivalent to the requirement that d_{ijk \cdots \upsilon} \in \{0, \pm x_{1}, \cdots , \pm x_{t} \} , where x_{1}, \cdots , x_{t} are commuting variables, and that \sum_{p} \sum_{q} \sum_{r} \cdots \sum_{y} d_{pqr \cdots ya} d_{pqr \cdots yb} = \left( \sum_{t} s_{i}x_{i}^{2} \right)^{g-1} \delta ab, where (s{1}, \cdots , s{t}) are integers giving the occurrences of \pm x_{1}, \cdots , \pm x_{t} in each row and column (this is called the type (s_{1}, \cdot ,s_{t})^{g-1}) and (pqr \cdots yz) represents all permutations of (ijk \cdots \upsilon) . This extends an idea of Paul J. Shlichta, whose higher dimensional Hadamard matrices are special cases with x_{1}, \cdots , x_{t} \in \{1,- 1\}, (s_{1}, \cdots, s_{t})=(g) , and (\sum_{t}s_{i}x_{i}^{2})=g . Another special case is higher dimensional weighing matrices of type (k)^{g} , which have x_{1}, \cdots , x_{t} \in \{0,1,- 1\}, (s_{1}, \cdots, s_{t})=(k) , and (\sum_{t}s_{i}x_{i}^{2})=k . Shlichta found proper g -dimensional Hadamard matrices of size (2^{t})^{g} . Proper orthogonal designs of type