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