Félix Gudiel

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.

GA Based Robust Blind Digital Watermarking

A genetic algorithm based robust blind digital watermarking scheme is presented. Starting from a binary image (the original watermark), a genetic algorithm is performed searching for a permutation of this image which is as uncorrelated as possible to the original watermark. The output of the GA is used as our final watermark, so that both security and robustness in the watermarking process is improved. Now, the original cover image is partitioned into non-overlapped square blocks (depending on the size of the watermark image). Then a (possibly extended) Hadamard transform is applied to these blocks, so that one bit information from the watermark image is embedded in each block by modifying the relationship of two coefficients in the transformed matrices. The watermarked image is finally obtained by simply performing the inverse (extended) Hadamard transform on the modified matrices. 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.

A Mixed Heuristic for Generating Cocyclic Hadamard Matrices

A way of generating cocyclic Hadamard matrices is described, which combines a new heuristic, coming from a novel notion of fitness, and a peculiar local search, defined as a constraint satisfaction problem. Calculations support the idea that finding a cocyclic Hadamard matrix of order

ACS searching for D 4t -Hadamard matrices

4 \cdot 47

On ZZt×ZZ22‐Cocyclic Hadamard Matrices

4·47 might be within reach, for the first time, progressing further upon the ideas explained in this work.

Embedding cocyclic D-optimal designs in cocyclic Hadamard matrices

Abstract Hadamard ideals were introduced in 2006 as a set of nonlinear polynomial equations whose zeros are uniquely related to Hadamard matrices with one or two circulant cores of a given order. Based on this idea, the cocyclic Hadamard test enables us to describe a polynomial ideal that characterizes the set of cocyclic Hadamard matrices over a fixed finite group G of order 4t. Nevertheless, the complexity of the computation of the reduced Grobner basis of this ideal is 2 O ( t 2 ) , which is excessive even for very small orders. In order to improve the efficiency of this polynomial method, we take advantage of some recent results on the inner structure of a cocyclic matrix to describe an alternative polynomial ideal that also characterizes the aforementioned set of cocyclic Hadamard matrices over G. The complexity of the computation decreases in this way to 2 O ( t ) . Particularly, we design two specific procedures for looking for Z t × Z 2 2 -cocyclic Hadamard matrices and D 4 t -cocyclic Hadamard matrices, so that larger cocyclic Hadamard matrices (up to t ≤ 39 ) are explicitly obtained.

On D4t‐Cocyclic 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 i- paths 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.