Hadamard matrices and their applications
Hadamard matrices and their applications
Product of four Hadamard matrices
Journal of Combinatorial Theory Series A
Williamson matrices of order 4n for n=33,35,39
Discrete Mathematics
Williamson matrices up to order 59
Designs, Codes and Cryptography
Analytic Combinatorics
Hadamard matrices and their applications: Progress 2007---2010
Cryptography and Communications
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Let S(x) be the number of n驴驴驴x for which a Hadamard matrix of order n exists. Hadamard's conjecture states that S(x) is about x/4. From Paley's constructions of Hadamard matrices, we have that $$ S(x) = \Omega\left( \frac{x}{\log x} \right). $$ In a recent paper, the first author suggested that counting the products of orders of Paley matrices would result in a greater density. In this paper we use results of Kevin Ford to show that it does: $$S(x) \geq \frac{x}{\log x} \exp\left((C+o(1))(\log \log \log x)^2 \right)\,, $$ where C驴=驴0.8178....This bound is surprisingly hard to improve upon. We show that taking into account all the other major known construction methods for Hadamard matrices does not shift the bound. Our arguments use the notion of a (multiplicative) monoid of natural numbers. We prove some initial results concerning these objects. Our techniques may be useful when assessing the status of other existence questions in design theory.