On $\mathbb{Z}_t \times \mathbb{Z}_2^2$-cocyclic Hadamard matrices

Abstract

A characterization of $\mathbb{Z} _t \times \mathbb{Z}_2^2$-cocyclic Hadamard matrices is described, depending on the notions of {\em distributions}, {\em ingredients} and {\em recipes}. In particular, these notions lead to the establishment of some bounds on the number and distribution of 2-coboundaries over $\mathbb{Z}_t \times \mathbb{Z} _2^2$ to use and the way in which they have to be combined in order to obtain a $\mathbb{Z} _t \times \mathbb{Z}_2^2$-cocyclic Hadamard matrix. Exhaustive searches have been performed, so that the table in p. 132 in [4] is corrected and completed. Furthermore, we identify four different operations on the set of coboundaries defining $\mathbb{Z} _t \times \mathbb{Z}_2^2$-cocyclic matrices, which preserve orthogonality. 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, in terms of {\em diagrams}. Let ${\cal H}$ be the set of cocyclic Hadamard matrices over $\mathbb{Z}_t \times \mathbb{Z}_2^2$ having a symmetric diagram. We also prove that the set of Williamson type matrices is a subset of ${\cal H}$ of size $\frac{|{\cal H}|}{t}$.