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.

Abstract. Given a basis B = {f1, . . . , f k } 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 Z Z2, 1 ≤ i ≤ k, whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that (x 1 , . . . , x k ) is a solution of the system if and only if the 2-cocycle f = f , 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 Z Z 2 2 × Z Z t and D4t. A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups D 4t , we have found that it suffices to check t instead of the 4t rows of M f , to decide the Hadamard character of the matrix (for a special class of cocycles f ).

In the literature, the theory of cocyclic Hadamard matrices has always been developed over finite groups. This paper introduces the natural generalization of this theory to be developed over Latin rectangles. In this regard, once we introduce the concept of binary cocycle over a given Latin rectangle, we expose examples of Hadamard matrices that are not cocyclic over finite groups but they are over Latin rectangles. Since it is also shown that not every Hadamard matrix is cocyclic over a Latin rectangle, we focus on answering both problems of existence of Hadamard matrices that are cocyclic over a given Latin rectangle and also its reciprocal, that is, the existence of Latin rectangles over which a given Hadamard matrix is cocyclic. We prove in particular that every Latin square over which a Hadamard matrix is cocyclic must be the multiplication table of a loop (not necessarily associative). Besides, we prove the existence of cocyclic Hadamard matrices over non-associative loops of order 2 t+3 , for all positive integer t > 0.

For a simplicial augmented algebra K, Eilenberg-Mac Lane constructed a chain map g B N K N → W N K . They proved that g is a reduction (homology isomorphism) and conjectured that it is also the injection of a contraction (special homotopy equivalence). The contraction C W−B W N K B N K N f g is followed at once by using homological perturbation techniques. If K is commutative, Eilenberg-Mac Lane proved that g is a morphism of DGAalgebras. The present article is devoted to proving that f and satisfy certain multiplicative properties (weaker than g) and showing how they can be used for computing in an economical way the homology of twisted cartesian products of two Eilenberg-Mac Lane spaces.

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.