The homological reduction method for computing cocyclic Hadamard matrices

Abstract

An alternate method for constructing (Hadamard) cocyclic matrices over a finite group G is described. Provided that a homological model φ:B (Z[G]) F H hG for G is known, the homological reduction method automatically generates a full basis for 2-cocycles over G (including 2-coboundaries). From these data, either an exhaustive or a heuristic search for Hadamard cocyclic matrices is then developed. The knowledge of an explicit basis for 2-cocycles which includes 2-coboundaries is a key point for the designing of the heuristic search. It is worth noting that some Hadamard cocyclic matrices have been obtained over groups G for which the exhaustive searching techniques are not feasible. From the computationalcost point of view, even in the case that the calculation of such a homological model is also included, comparison with other methods in the literature shows that the homological reduction method drastically reduces the required computing time of the operations involved, so that even exhaustive searches succeeded at orders for which previous calculations could not be completed. With aid of an implementation of the method in Mathematica, some examples are discussed, including the case of very well-known groups (finite abelian groups, dihedral groups) for clarity.

Homological models for semidirect products of finitely generated Abelian groups

“…The third approach to this question, which we term the homological reduction method, is described in [4]. Provided a homological model hG for G is known (that is, a differential graded module of finite type which shares the homology groups with G), it explicitly describes an algorithm for constructing a basis for 2-cocycles over G in a straightforward manner.…”

“…In fact, the goodness of this approach is supported by the efficiency in which both H 1 (G) G/ [G, G] and H 2 (G) are computed from the homological model hG. In [5], the cohomological analogous to this method is described and applied for computing n-cocycles in general. It might be a potential source of examples for cocyclic matrices of higher dimensions, which may not be supplied by the other methods.…”

“…An extended version of the method for iterated products of central extensions and semidirect products of finitely generated Abelian groups has been implemented in Mathematica by the authors (see [1,3]). Some calculations with this package have led to the finding new cocyclic Hadamard matrices [2,4].…”

“…Using only the projection f and the differential d described in the example above, a generating set of representative 2-cocycles and 3-cocycles over D 4t are given in [4,5], respectively. These computations have led to the finding of new cocyclic Hadamard matrices [4, Table 1].…”

“…We provide a method for constructing an explicit contraction (special homotopy equivalence) from the reduced bar construction of the group ring of G, B(Z Z[G]), to a much smaller DGA-module hG. Such a contraction is called a homological model for G and is used as the input datum in the methods described in Álvarez et al (J Symb Comput 44:558-570, 2009;2012) for calculating a generating set for representative 2-cocycles and n-cocycles over G, respectively. These computations have led to the finding of new cocyclic Hadamard matrices (Álvarez et al in 2006).…”