Some new operations on Zt x Z2,2-cocyclic Hadamard matrices

TL;DR

This paper introduces diagrammatic tools and four operations on Zt x Z2,2-cocyclic Hadamard matrices, enabling the generation of new matrices and efficient classification into disjoint orbits.

Contribution

It presents new diagrammatic representations and operations that preserve Hadamard properties, facilitating the construction and classification of cocyclic Hadamard matrices.

Findings

Defined diagrams for visualizing coboundaries

Introduced four operations preserving Hadamard matrices

Simplified classification into disjoint orbits

Abstract

Following the ideas of [AGG11] about Zt x Z2,2-cocyclic Hadamard matrices, we introduce the notion of diagram, which visually represents any set of coboundaries. Diagrams are a very useful tool for the description and the study of paths and intersections, as described in [AGG11]. Then, we will study four different operations on Zt x Z2,2-cocyclic matrices. These operations will be defined on the set of coboundaries defining the matrix, preserve the Hadamard character of the cocyclic matrices, and allow us to obtain new Hadamard matrices from old ones. We split the set of Hadamard matrices into disjoint orbits, define representatives for them and take advantage of this fact to compute them in an easier way than the usual purely exhaustive way.