Williamson Matrices and a Conjecture of Ito‘s

• Authors: Bernhard Schmidt
• Venue: Designs, Codes and Cryptography - Special issue on designs and codes—a memorial tribute to Ed Assmus
• Year: 1999

We point out an interesting connectionbetween Williamson matrices and relative difference sets in nonabeliangroups. As a consequence, we are able to show that there arerelative (4t,2,4t,2t)-difference sets in the dicyclicgroups Q_{8t}=\la a,b|a^{4t}=b^4=1, a^{2t}=b^2, b^{-1}ab=a^{-1}\rafor all t of the form t=2^a\cdot 10^b \cdot26^c \cdot m with a,b,c\ge 0, m\equiv1\ (\mod 2), whenever 2m-1 or 4m-1is a prime power or there is a Williamson matrix over \Z_m.This gives further support to an important conjecture of ItoIT5 which asserts that there are relative (4t,2,4t,2t)-differencesets 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-1is 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.