Williamson matrices and a conjecture of Ito's

We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t,2,4t,2t)-difference sets in the dicyclic groups Q_{8t}=\la a,b|a^{4t}=b^4=1, a^{2t}=b^2, b^{-1}ab=a^{-1}\ra for all...

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Main Author: Bernhard, Schmidt.
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2009
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Online Access:https://hdl.handle.net/10356/92273
http://hdl.handle.net/10220/6030
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Institution: Nanyang Technological University
Language: English
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spelling sg-ntu-dr.10356-922732023-02-28T19:35:33Z Williamson matrices and a conjecture of Ito's Bernhard, Schmidt. School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t,2,4t,2t)-difference sets in the dicyclic groups Q_{8t}=\la a,b|a^{4t}=b^4=1, a^{2t}=b^2, b^{-1}ab=a^{-1}\ra for all t of the form t=2^a\cdot 10^b \cdot 26^c \cdot m with a,b,c\ge 0, m\equiv 1\ (\mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over \Z_m. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t,2,4t,2t)-difference sets in Q_{8t} for every positive integer t. We also give simpler alternative constructions for relative (4t,2,4t,2t) -difference sets in Q_{8t} for all t such that 2t-1 or 4t-1 is a prime power. Relative difference sets in Q_{8t} with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito‘s conjecture for all t\le 46. Accepted version 2009-08-11T03:07:57Z 2019-12-06T18:20:27Z 2009-08-11T03:07:57Z 2019-12-06T18:20:27Z 1999 1999 Journal Article Bernhard, S. (1999). Williamson matrices and a conjecture of Ito's. Journal of designs codes and cryptography, 17(1-3), 61-68. 0925-1022 https://hdl.handle.net/10356/92273 http://hdl.handle.net/10220/6030 10.1023/A:1008398319853 en Journal of designs codes and cryptography. Designs codes and cryptography © copyright 1999 Springer Netherlands. The journal's website is located at http://www.springerlink.com/content/m70j6m607k1630g2. 11 p. application/pdf
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics
spellingShingle DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics
Bernhard, Schmidt.
Williamson matrices and a conjecture of Ito's
description We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t,2,4t,2t)-difference sets in the dicyclic groups Q_{8t}=\la a,b|a^{4t}=b^4=1, a^{2t}=b^2, b^{-1}ab=a^{-1}\ra for all t of the form t=2^a\cdot 10^b \cdot 26^c \cdot m with a,b,c\ge 0, m\equiv 1\ (\mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over \Z_m. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t,2,4t,2t)-difference sets in Q_{8t} for every positive integer t. We also give simpler alternative constructions for relative (4t,2,4t,2t) -difference sets in Q_{8t} for all t such that 2t-1 or 4t-1 is a prime power. Relative difference sets in Q_{8t} with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito‘s conjecture for all t\le 46.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Bernhard, Schmidt.
format Article
author Bernhard, Schmidt.
author_sort Bernhard, Schmidt.
title Williamson matrices and a conjecture of Ito's
title_short Williamson matrices and a conjecture of Ito's
title_full Williamson matrices and a conjecture of Ito's
title_fullStr Williamson matrices and a conjecture of Ito's
title_full_unstemmed Williamson matrices and a conjecture of Ito's
title_sort williamson matrices and a conjecture of ito's
publishDate 2009
url https://hdl.handle.net/10356/92273
http://hdl.handle.net/10220/6030
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