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In this paper, we describe some necessary and sufficient conditions for a set of coboundaries to yield a cocyclic Hadamard matrix over the dihedral group [ABSTRACT FROM AUTHOR]

Provided that a cohomological model for $$G$$ is known, we describe a method for constructing a basis for $$n$$ -cocycles over $$G$$ , from which the whole set of $$n$$ -dimensional $$n$$ -cocyclic matrices over $$G$$ may be straightforwardly calculated. Focusing in the case $$n=2$$ (which is of special interest, e.g. for looking for cocyclic Hadamard matrices), this method provides a basis for 2-cocycles in such a way that representative $$2$$ -cocycles are calculated all at once, so that there is no need to distinguish between inflation and transgression 2-cocycles (as it has traditionally been the case until now). When $$n>2$$ , this method provides an uniform way of looking for higher dimensional n-cocyclic Hadamard matrices for the first time. We illustrate the method with some examples, for $$n=2,3$$ . In particular, we give some examples of improper 3-dimensional $$3$$ -cocyclic Hadamard matrices. [ABSTRACT FROM AUTHOR]

Abstract: An alternate method for constructing (Hadamard) cocyclic matrices over a finite group is described. Provided that a homological model for is known, the homological reduction method automatically generates a full basis for 2-cocycles over (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 for which the exhaustive searching techniques are not feasible. From the computational-cost 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. [Copyright &y& Elsevier]

Since Horadam and de Launey introduced the cocyclic framework on combinatorial designs in the 1990s, it has revealed itself as a powerful technique for looking for (cocyclic) Hadamard matrices. Ten years later, the series of papers by Kotsireas, Koukouvinos and Seberry about Hadamard matrices with one or two circulant cores introduced a different structured approach to the Hadamard conjecture. This paper is built on both strengths, so that Hadamard matrices with cocyclic cores are introduced and studied. They are proved to strictly include usual Hadamard matrices with one and two circulant cores, and therefore provide a wiser uniform approach to a structured Hadamard conjecture. [ABSTRACT FROM AUTHOR]

About twenty-five years ago, Horadam and de Launey introduced the cocyclic development of designs, from which the notion of cocyclic Hadamard matrices developed over a group was readily derived. Much more recently, it has been proved that this notion may naturally be extended to define cocyclic Hadamard matrices developed over a loop. This paper delves into this last topic by introducing the concepts of coboundary, pseudocoboundary and pseudococycle over a quasigroup, and also the notion of the pseudococyclic Hadamard matrix. Furthermore, Goethals-Seidel loops are introduced as a family of Moufang loops so that every Hadamard matrix of Goethals-Seidel type (which is known not to be cocyclically developed over any group) is actually pseudococyclically developed over them. Finally, we also prove that, no matter if they are pseudococyclic matrices, the usual cocyclic Hadamard test is unexpectedly applicable. [ABSTRACT FROM AUTHOR]