Construction of Proper Higher Dimensional Hadamard Matrices from Perfect Binary Arrays

Abstract

We describe several techniques for constructing $n$-dimensional Hadamard matrices from $2$-dimensional Hadamard matrices, and note that they may be applied to any perfect binary array $(PBA)$, thus optimally improving a result of Yang. We introduce cocyclic perfect binary arrays, whose energy is not restricted to being a perfect square. These include
all of Jedwab’s generalized perfect binary arrays. There are many more cocyclic $PBAs$ than $PBAs$. We resolve a potential ambiguity inherent in the “weak difference set” construction of $n$-dimensional Hadamard matrices from cocyclic $PBAs$ and show it
is a relative difference set construction.