arXiv subjects

Félix Gudiel

Publications and source records attributed to Félix Gudiel.

4 recordsLinked to original sources

arXiv2026

Fundamentals of the cocyclic development of Hadamard matrices over loops

This paper establishes a formal framework for the cocyclic development of Hadamard matrices over loops based on a novel cohomology theory with associativity obstructions. In this context, we formalize the notion of a loop-cocyclic Hadamard matrix, so that both the classical cocyclic Hadamard matrices over groups and the recently introduced pseudococyclic Hadamard matrices over loops are naturally embedded within this broader algebraic architecture. To validate the computational viability of this framework, we prove the existence of four new Hadamard equivalence classes of order 24 and eight new classes of order 28 that are strictly non-cocyclic over any finite group, arising uniquely as loop-cocyclic developments over Moufang and right Bol loops.

math.CO↗

arXiv2012

Embedding cocyclic D-optimal designs in cocyclic Hadamard matrices

In this paper 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.

math.CO↗

arXiv2012

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.

math.CO↗

arXiv2012

Searching for partial Hadamard matrices

Three algorithms looking for pretty large partial Hadamard matrices are described. Here "large" means that hopefully about a third of a Hadamard matrix (which is the best asymptotic result known so far, [dLa00]) 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 [SG06]. 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 G_t of Ito's Hadamard Graph Delta (4t) [Ito85], since cliques of order m in G_t can be seen as (m+3)*4t partial Hadamard matrices.

math.CO↗