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      Williamson matrices and a conjecture of Ito's.

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      Accepted version (181.2Kb)
      Author
      Bernhard, Schmidt.
      Date of Issue
      1999
      School
      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.
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      http://dx.doi.org/10.1023/A:1008398319853
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