36 research outputs found
Self-Dual codes from (−1,1)-matrices of skew symmetric type
Previously, self-dual codes have been constructed from weighing matrices,
and in particular from conference matrices (skew and symmetric). In this paper, codes
constructed from matrices of skew symmetric type whose determinants reach the Ehlich-
Wojtas’ bound are presented. A necessary and sufficient condition for these codes to be
self-dual is given, and examples are provided for lengths up to 52.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM-016Junta de Andalucía P07-FQM-0298
A Heuristic Procedure with Guided Reproduction for Constructing Cocyclic Hadamard Matrices
A genetic algorithm for constructing cocyclic Hadamard matrices
over a given group is described. The novelty of this algorithm is
the guided heuristic procedure for reproduction, instead of the classical
crossover and mutation operators. We include some runs of the algorithm
for dihedral groups, which are known to give rise to a large amount of
cocyclic Hadamard matrices.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM–296Junta de Andalucía P07-FQM-0298
Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach
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.Junta de Andalucía FQM-01
Error correcting codes from quasi-Hadamard matrices
Levenshtein described in [5] a method for constructing error
correcting codes which meet the Plotkin bounds, provided suitable Ha-
damard matrices exist. Uncertainty about the existence of Hadamard
matrices on all orders multiple of 4 is a source of difficulties for the prac-
tical application of this method. Here we extend the method to the case
of quasi-Hadamard matrices. Since efficient algorithms for constructing
quasi-Hadamard matrices are potentially available from the literature
(e.g. [7]), good error correcting codes may be constructed in practise.
We illustrate the method with some examples.Junta de Andalucía FQM–29
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.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM-016Junta de Andalucía P07-FQM-0298
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 D4t 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.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM–296Junta de Andalucía P07-FQM-0298
Searching for partial Hadamard matrices
Three algorithms looking for pretty large partial Hadamard ma-
trices are described. Here “large” means that hopefully about a third of a
Hadamard matrix (which is the best asymptotic result known so far, [8]) is
achieved. The first one performs some kind of local exhaustive search, and
consequently is expensive from the time consuming point of view. The second
one comes from the adaptation of the best genetic algorithm known so far
searching for cliques in a graph, due to Singh and Gupta [21]. The last one
consists in another heuristic search, which prioritizes the required processing
time better than the final size of the partial Hadamard matrix to be obtained. In
all cases, the key idea is characterizing the adjacency properties of vertices in a
particular subgraph Gt of Ito’s Hadamard Graph (4t) [18], since cliques of
order m in Gt can be seen as (m + 3) × 4t partial Hadamard matrices.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM-016Junta de Andalucía P07-FQM-0298
ACS Searching for D4t-Hadamard Matrices
An Ant Colony System (ACS) looking for cocyclic Hadamard
matrices over dihedral groups D4t is described. The underlying weighted
graph consists of the rooted trees described in [1], whose vertices are
certain subsets of coboundaries. A branch of these trees defines a D4t-
Hadamard matrix if and only if two conditions hold: (i) Ii = i − 1 and,
(ii) ci = t, for every 2 ≤ i ≤ t, where Ii and ci denote the number of ipaths
and i-intersections (see [3] for details) related to the coboundaries
defining the branch. The pheromone and heuristic values of our ACS are
defined in such a way that condition (i) is always satisfied, and condition
(ii) is closely to be satisfied.Ministerio de Ciencia e Innovación MTM2008-06578Junta de Andalucía FQM–296Junta de Andalucía P07-FQM-0298
GA Based Robust Blind Digital Watermarking
A genetic algorithm based robust blind digital watermarking scheme is presented.
The experimental results show that our scheme keeps invisibility, security and robustness
more likely than other proposals in the literature, thanks to the GA pretreatment.Junta de Andalucía FQM-01
Homological models for semidirect products of finitely generated Abelian groups
Let G be a semidirect product of finitely generated Abelian groups. We provide a method for constructing an explicit contraction (special homotopy equivalence) from the reduced bar construction of the group ring of G, B¯¯¯¯(ZZ[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)