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Construction of proper higher dimensional Hadamard matrices from perfect binary arrays. (English) Zbl 0917.05016
Summary: 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 generalised 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.

##### MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
##### Keywords:
Hadamard matrices; perfect binary array; difference set