13 research outputs found
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
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
COMPLEX HADAMARD MATRICES AND APPLICATIONS
A complex Hadamard matrix is a square matrix H â M N (C) whose entries are on the unit circle, |H ij | = 1, and whose rows and pairwise orthogonal. The main example is the Fourier matrix, F N = (w ij) with w = e 2Ïi/N. We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects. CONTENT