José Andrés Armario

The homological reduction method for computing cocyclic Hadamard matrices

An alternate method for constructing (Hadamard) cocyclic matrices over a finite group G is described. Provided that a homological model 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 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.

A genetic algorithm for cocyclic hadamard matrices

A genetic algorithm for finding cocyclic Hadamard matrices is described. Though we focus on the case of dihedral groups, the algorithm may be easily extended to cover any group. Some executions and examples are also included, with aid of Mathematica 4.0.

Calculating cocyclic hadamard matrices in mathematica : exhaustive and heuristic searches

We describe a notebook in Mathematica which, taking as input data a homological model for a finite group G of order |G| = 4t, performs an exhaustive search for constructing the whole set of cocyclic Hadamard matrices over G. Since such an exhaustive search is not practical for orders 4t ≥28, the program also provides an alternate method, in which an heuristic search (in terms of a genetic algorithm) is performed. We include some executions and examples.

A mathematica notebook for computing the homology of iterated products of groups

Let G be a group which admits the structure of an iterated product of central extensions and semidirect products of abelian groups Gi (both finite and infinite). We describe a Mathematica 4.0 notebook for computing the homology of G, in terms of some homological models for the factor groups Gi and the products involved. Computational results provided by our program have allowed the simplification of some of the formulae involved in the calculation of Hn(G). Consequently the efficiency of the method has been improved as well. We include some executions and examples.

Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach

Abstract An n by n skew-symmetric type (-1; 1)-matrix K =[ki;j ] has 1’s on the main diagonal and ±1’s elsewhere with ki;j =-kj;i . The largest possible determinant of such a matrix K is an interesting problem. The literature is extensive for n ≡ 0 mod 4 (skew-Hadamard matrices), but for n ≡ 2 mod 4 there are few results known for this question. In this paper we approach this problem constructing cocyclic matrices over the dihedral group of 2t elements, for t odd, which are equivalent to (-1; 1)-matrices of skew type. Some explicit calculations have been done up to t =11. To our knowledge, the upper bounds on the maximal determinant in orders 18 and 22 have been improved.

Rooted Trees Searching for Cocyclic Hadamard Matrices over D4t

A new reduction on the size of the search space for cocyclic Hadamard matrices over dihedral groups D 4t is described, in terms of the so called central distribution . This new search space adopt the form of a forest consisting of two rooted trees (the vertices representing subsets of coboundaries) which contains all cocyclic Hadamard matrices satisfying the constraining condition. Experimental calculations indicate that the ratio between the number of constrained cocyclic Hadamard matrices and the size of the constrained search space is greater than the usual ratio.

An Algorithm for Computing Cocyclic Matrices Developed over Some Semidirect Products

An algorithm for calculating a set of generators of representative 2-cocycles on semidirect product offinite abelian groups is constructed, in light of the theory over cocyclic matrices developed by Horadam and de Launey in [7,8]. The method involves some homological perturbation techniques [3,1], in the homological correspondent to the work which Grabmeier and Lambe described in [12] from the viewpoint of cohomology. Examples of explicit computations over all dihedral groups D4t are given, with aid of MATHEMATICA.

On (−1, 1)-Matrices of Skew Type with the Maximal Determinant and Tournaments

Skew Hadamard matrices of order n give the solution to the question of finding the largest possible n by n determinant with entries ± 1 of skew type when \(n \equiv 0\pmod 4\). Characterizations of skew Hadamard matrices in terms of tournaments are well-known. For \(n \equiv 2\pmod 4\), we give a characterization of (−1, 1)-matrices of skew type of order n where their determinants reach Ehlich–Wojtas’ bound in terms of tournaments.

On an inequivalence criterion for cocyclic Hadamard matrices

Given two Hadamard matrices of the same order, it can be quite difficult to decide whether or not they are equivalent. There are some criteria to determine Hadamard inequivalence. Among them, one of the most commonly used is the 4-profile criterion. In this paper, a reformulation of this criterion in the cocyclic framework is given. The improvements obtained in the computation of the 4-profile of a cocyclic Hadamard matrix are indicated.

Embedding cocylic D-optimal designs in cocylic Hadamard matrices

A method for embedding cocyclic submatrices with “large” determinants of orders 2t in certain cocyclic Hadamard matrices of orders 4t is described (t an odd integer). If these determinants attain the largest possible value, we are embedding D-optimal designs. Applications to the pivot values that appear when Gaussian elimination with complete pivoting is performed on these cocyclic Hadamard matrices are studied.