18 research outputs found
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
On the Asymptotic Existence of Hadamard Matrices
It is conjectured that Hadamard matrices exist for all orders ().
However, despite a sustained effort over more than five decades, the strongest
overall existence results are asymptotic results of the form: for all odd
natural numbers , there is a Hadamard matrix of order ,
where and are fixed non-negative constants. To prove the Hadamard
Conjecture, it is sufficient to show that we may take and . Since
Seberry's ground-breaking result, which showed that we may take and
, there have been several improvements where has been by stages
reduced to 3/8. In this paper, we show that for all , the set of
odd numbers for which there is a Hadamard matrix of order
has positive density in the set of natural numbers.
The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that
there are enough Paley Hadamard matrices to give the result.Comment: Keywords: Hadamard matrices, Asymptotic existence, Cocyclic Hadamard
matrices, Relative difference sets, Riesel numbers, Extended Riemann
hypothesis. (Received 2 August 2008, Available online 18 March 2009
Cohomology-Developed Matrices -- constructing families of weighing matrices and automorphism actions
The aim of this work is to construct families of weighing matrices via their
automorphism group action. This action is determined from the
-cohomology groups of the underlying abstract group. As a consequence,
some old and new families of weighing matrices are constructed. These include
the Paley Conference, the Projective-Space, the Grassmannian, and the
Flag-Variety weighing matrices. We develop a general theory relying on low
dimensional group-cohomology for constructing automorphism group actions, and
in turn obtain structured matrices that we call \emph{Cohomology-Developed
matrices}. This "Cohomology-Development" generalizes the Cocyclic and Group
Developments. The Algebraic structure of modules of Cohomology-Developed
matrices is discussed, and an orthogonality result is deduced. We also use this
algebraic structure to define the notion of \emph{Quasiproducts}, which is a
generalization of the Kronecker-product
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
The cohomological reduction method for computing n-dimensional cocyclic matrices
Provided that a cohomological model for is known, we describe a method
for constructing a basis for -cocycles over , from which the whole set of
-dimensional -cocyclic matrices over may be straightforwardly
calculated. Focusing in the case (which is of special interest, e.g. for
looking for cocyclic Hadamard matrices), this method provides a basis for
2-cocycles in such a way that representative -cocycles are calculated all at
once, so that there is no need to distinguish between inflation and
transgression 2-cocycles (as it has traditionally been the case until now).
When , this method provides an uniform way of looking for higher
dimensional -cocyclic Hadamard matrices for the first time. We illustrate
the method with some examples, for . In particular, we give some
examples of improper 3-dimensional -cocyclic Hadamard matrices.Comment: 17 pages, 0 figure
Cocyclic Hadamard Matrices: An Efficient Search Based Algorithm
This dissertation serves as the culmination of three papers. âCounting the decimation classes of binary vectors with relatively prime fixed-density presents the first non-exhaustive decimation class counting algorithm. âA Novel Approach to Relatively Prime Fixed Density Bracelet Generation in Constant Amortized Time presents a novel lexicon for binary vectors based upon the Discrete Fourier Transform, and develops a bracelet generation method based upon the same. âA Novel Legendre Pair Generation Algorithm expands upon the bracelet generation algorithm and includes additional constraints imposed by Legendre Pairs. It further presents an efficient sorting and comparison algorithm based upon symmetric functions, as well as multiple unique Legendre Pairs