930 research outputs found
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
Equivalences of Zt×Z22-cocyclic Hadamard matrices
One of the most promising structural approaches to resolving the
Hadamard Conjecture uses the family of cocyclic matrices over Zt × Z2
2.
Two types of equivalence relations for classifying cocyclic matrices over
Zt × 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 × 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 “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
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 − xlij
: (i, j) is a non-empty cell inL ⊆ 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
On the computability of the p-local homology of twisted cartesian products of Eilenberg-Mac Lane spaces
Working in the framework of the Simplicial Topology, a method for calculating
the p-local homology of a twisted cartesian product X( , m, , 0, n) =
K( ,m)× K( 0, n) of Eilenberg-Mac Lane spaces is given. The chief technique
is the construction of an explicit homotopy equivalence between the normalized
chain complex of X and a free DGA-module of finite type M, via homological
perturbation. If X is a commutative simplicial group (being its inner product
the natural one of the cartesian product of K( ,m) and K( 0, n)), then M is a
DGA-algebra. Finally, in the special case K( , 1) ,! X
p!
K( 0, n), we prove
that M can be a small twisted tensor product
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
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
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)
An algorithm for computing cocyclic matrices developed over some semidirect products
An algorithm for calculating a set ofgenerators ofrepresentative 2-cocycles on semidirect product offinite abelian groups is constructed, in light ofthe theory over cocyclic matrices developed by Horadam and de Launey in [7],[8]. The method involves some homological perturbation techniques [3],[1], in the homological correspondent to the work which Grabmeier and Lambe described in [12] from the viewpoint ofcohomology . Examples ofexplicit computations over all dihedral groups D 4t are given, with aid of Mathematica
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