NDLI: On the density of the set of known Hadamard orders

Content Provider	Springer Nature Link
Author	Launey, Warwick Gordon, Daniel M.
Copyright Year	2010
Abstract	Let S(x) be the number of n ≤ 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 = 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.
Starting Page	233
Ending Page	246
Page Count	14
File Format	PDF
ISSN	19362447
Journal	Cryptography and Communications
Volume Number	2
Issue Number	2
e-ISSN	19362455
Language	English
Publisher	Springer US
Publisher Date	2010-05-18
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Hadamard matrices Paley matrices Asymptotic density of sets Multiplicative functions Mathematics of Computing Information and Communication, Circuits Communications Engineering, Networks Coding and Information Theory Data Structures, Cryptology and Information Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computer Networks and Communications Computational Theory and Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Similar Documents

Some REALLY beautiful Hadamard matrices

Unbiased complex Hadamard matrices and bases

On an inequivalence criterion for cocyclic Hadamard matrices

A Fourier-analytic approach to counting partial Hadamard matrices

Hadamard matrices and their applications: Progress 2007–2010

Some remarks on Hadamard matrices

Exotic complex Hadamard matrices and their equivalence

Block weighing matrices

On twin prime power Hadamard matrices

On the density of the set of known Hadamard orders