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# Williamson matrices and a conjecture of Ito's.

 dc.contributor.author Bernhard, Schmidt. dc.date.accessioned 2009-08-11T03:07:57Z dc.date.available 2009-08-11T03:07:57Z dc.date.copyright 1999 dc.date.issued 2009-08-11T03:07:57Z dc.identifier.citation Bernhard, S. (1999). Williamson matrices and a conjecture of Ito's. Journal of designs codes and cryptography, 17(1-3), 61-68. dc.identifier.issn 0925-1022 dc.identifier.uri http://hdl.handle.net/10220/6030 dc.description.abstract 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. dc.format.extent 11 p. dc.language.iso en dc.relation.ispartofseries Journal of designs codes and cryptography. dc.rights Designs codes and cryptography © copyright 1999 Springer Netherlands. The journal's website is located at http://www.springerlink.com/content/m70j6m607k1630g2. dc.subject DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics. dc.title Williamson matrices and a conjecture of Ito's. dc.type Journal Article dc.contributor.school School of Physical and Mathematical Sciences dc.identifier.doi http://dx.doi.org/10.1023/A:1008398319853 dc.description.version Accepted version

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