A system of equations for describing cocyclic Hadamard matrices

Abstract. Given a basis B = {f1,..., fk} for 2-cocycles f: G×G → {±1} over a group G of order |G | = 4t, we describe a non-linear system of 4t − 1 equations and k indeterminates xi over ZZ2, 1 ≤ i ≤ k, whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that (x1,..., xk) is a solution of the system if and only if the 2-cocycle f = fx11 · · · fxkk gives rise to a cocyclic Hadamard matrix Mf = (f(gi, gj)). Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in B 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 ZZ22×ZZt and D4t. A deeper study of the system provides some more nice properties. For in-stance, in the case of dihedral groups D4t, we have found that it suffices to check t instead of the 4t rows of Mf, to decide the Hadamard character of the matrix (for a special class of cocycles f).