22 research outputs found
Determinants of (â1,1)-matrices of the skew-symmetric type: a cocyclic approach
An n by n skew-symmetric type (â1, 1)-matrix K = [ki,j ] has 1âs on the main
diagonal and ±1âs elsewhere with ki,j = âkj,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 (â1, 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 AndalucĂa FQM-01
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
Embedding cocylic D-optimal designs in cocylic Hadamard matrices
A method for embedding cocyclic submatrices with âlargeâ 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 InnovaciĂłn MTM2008-06578Junta de AndalucĂa FQM-016Junta de AndalucĂa P07-FQM-0298
A computational algebraic geometry approach to analyze pseudo-random sequences based on Latin squares
Latin squares are used as scramblers on symmetric-key algorithms that generate
pseudo-random sequences of the same length. The robustness and effectiveness of
these algorithms are respectively based on the extremely large key space and the
appropriate choice of the Latin square under consideration. It is also known the
importance that isomorphism classes of Latin squares have to design an effective
algorithm. In order to delve into this last aspect, we improve in this paper the efficiency
of the known methods on computational algebraic geometry to enumerate and
classify partial Latin squares. Particularly, we introduce the notion of affine algebraic
set of a partial Latin square L = (lij ) of order n over a field K as the set of zeros
of the binomial ideal xi xj â xlij
: (i, j) is a non-empty cell inL â K[x1, . . . , xn].
Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets,
every isomorphism invariant of the latter constitutes an isomorphism invariant of the
former. In particular, we deal computationally with the problem of deciding whether
two given partial Latin squares have either the same or isomorphic affine algebraic
sets. To this end, we introduce a new pair of equivalence relations among partial
Latin squares: being partial transpose and being partial isotopic
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 InnovaciĂłn MTM2008-06578Junta de AndalucĂa FQMâ296Junta de AndalucĂa P07-FQM-0298
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
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