Equivalences of Zt×Z22-cocyclic Hadamard matrices

Abstract

One of the most promising structural approaches to resolving the Hadamard Conjecture uses the family of cocyclic matrices over Zt × Z2 2. Two types of equivalence relations for classifying cocyclic matrices over Zt × Z2 2 have been found. Any cocyclic matrix equivalent by either of these relations to a Hadamard matrix will also be Hadamard. One type, based on algebraic relations between cocycles over any fi- nite group, has been known for some time. Recently, and independently, a second type, based on four geometric relations between diagrammatic visualisations of cocyclic matrices over Zt × Z2 2, has been found. Here we translate the algebraic equivalences to diagrammatic equivalences and show one of the diagrammatic equivalences cannot be obtained this way. This additional equivalence is shown to be the geometric translation of matrix transposition.