A System of Equations for Describing Cocyclic Hadamard Matrices

Abstract

Given a basis equation image for 2-cocycles equation image over a group G of order equation image, we describe a nonlinear system of 4t-1 equations and k indeterminates equation image over equation image, whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that (equation image) is a solution of the system if and only if the 2-cocycle equation image gives rise to a cocyclic Hadamard matrix equation image. Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in equation image which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups equation image and equation image. A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups equation image, we have found that it suffices to check t instead of the 4t rows of equation image, to decide the Hadamard character of the matrix (for a special class of cocycles f).