Williamson matrices and a conjecture of Ito's.
Author
Bernhard, Schmidt.
Date of Issue
1999School
School of Physical and Mathematical Sciences
Version
Accepted version
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.
Subject
DRNTU::Science::Mathematics::Discrete mathematics::Combinatorics
Type
Journal Article
Series/Journal Title
Journal of designs codes and cryptography.
Rights
Designs codes and cryptography © copyright 1999 Springer Netherlands. The journal's website is located at http://www.springerlink.com/content/m70j6m607k1630g2.
Collections
http://dx.doi.org/10.1023/A:1008398319853
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