15 research outputs found
Equivalences of ZtĂZ22-cocyclic Hadamard matrices
One of the most promising structural approaches to resolving the
Hadamard Conjecture uses the family of cocyclic matrices over Zt Ă Z2
2.
Two types of equivalence relations for classifying cocyclic matrices over
Zt Ă Z2
2 have been found. Any cocyclic matrix equivalent by either of
these relations to a Hadamard matrix will also be Hadamard.
One type, based on algebraic relations between cocycles over any fi-
nite group, has been known for some time. Recently, and independently,
a second type, based on four geometric relations between diagrammatic
visualisations of cocyclic matrices over Zt Ă Z2
2, has been found. Here
we translate the algebraic equivalences to diagrammatic equivalences and
show one of the diagrammatic equivalences cannot be obtained this way.
This additional equivalence is shown to be the geometric translation of
matrix transposition
Cocyclic Hadamard matrices over Latin rectangles
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 2t+3, for all positive integer t > 0.Junta de AndalucĂa FQM-01
Searching for partial Hadamard matrices
Three algorithms looking for pretty large partial Hadamard ma-
trices are described. Here âlargeâ 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 Itoâs Hadamard Graph (4t) [18], since cliques of
order m in Gt can be seen as (m + 3) Ă 4t partial Hadamard matrices.Ministerio de Ciencia e InnovaciĂłn MTM2008-06578Junta de AndalucĂa FQM-016Junta de AndalucĂa 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 â 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 InnovaciĂłn MTM2008-06578Junta de AndalucĂa FQMâ296Junta de AndalucĂa 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 AndalucĂa FQM-01
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 4 · 47 might be within reach, for
the first time, progressing further upon the ideas explained in this work.Junta de AndalucĂa FQM-01
Generating binary partial Hadamard matrices
This paper deals with partial binary Hadamard matrices. Although there is a fast simple
way to generate about a half (which is the best asymptotic bound known so far, see de
Launey (2000) and de Launey and Gordon (2001)) of a full Hadamard matrix, it cannot
provide larger partial Hadamard matrices beyond this bound. In order to overcome such
a limitation, we introduce a particular subgraph Gt of Itoâs Hadamard Graph Î(4t) (Ito,
1985), and study some of its properties,which facilitates that a procedure may be designed
for constructing large partial Hadamard matrices. The key idea is translating the problem
of extending a given clique in Gt into a Constraint Satisfaction Problem, to be solved
by Minion (Gent et al., 2006). Actually, iteration of this process ends with large partial
Hadamard matrices, usually beyond the bound of half a full Hadamard matrix, at least as
our computation capabilities have led us thus far
On Cocyclic Hadamard Matrices over Goethals-Seidel Loops
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.Junta de AndalucĂa FQM-01
Generating partial Hadamard matrices as solutions to a Constraint Satisfaction Problem characterizing cliques
A procedure is described looking for partial Hadamard matrices, as cliques of a particular subgraph Gt of Itoâs Hadamard Graph Î(4t) [9]. The key idea is translating the problem of extending a given clique Cm to a larger clique of size m+ 1 in Gt, into a constraint satisfaction problem, and look for a solution to this problem by means of Minion [6]. Iteration of this process usually ends with a large partial Hadamard matrix.Junta de AndalucĂa FMQ-01