%0 Journal Article %1 schmidt99 %A Schmidt, Bernhard %D 1999 %J Designs, Codes and Cryptography %K hadamard williamson %N 1 %P 61--68 %R 10.1023/A:1008398319853 %T Williamson Matrices and a Conjecture of Ito's %V 17 %X 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 Q8t = 〈a, b|a4t = b4 = 1, a2t = b2, b-1ab = a-1〉 for all t of the form t = 2a · 10 b · 26 c · m with a, b, c ≥ 0, m ≡ 1\backslash (mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over ℤm. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t, 2, 4t, 2t)-difference sets in Q8t for every positive integer t. We also give simpler alternative constructions for relative (4t, 2, 4t, 2t)-difference sets in Q8t for all t such that 2t - 1 or 4t - 1 is a prime power. Relative difference sets in Q8t with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito's conjecture for all t ≤ 46.

@article{schmidt99, 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 Q8t = {\textlangle}a, b|a4t = b4 = 1, a2t = b2, b-1ab = a-1{\textrangle} for all t of the form t = 2a {\textperiodcentered} 10 b {\textperiodcentered} 26 c {\textperiodcentered} m with a, b, c ≥ 0, m ≡ 1{\backslash} (mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over ℤm. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t, 2, 4t, 2t)-difference sets in Q8t for every positive integer t. We also give simpler alternative constructions for relative (4t, 2, 4t, 2t)-difference sets in Q8t for all t such that 2t - 1 or 4t - 1 is a prime power. Relative difference sets in Q8t with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito's conjecture for all t ≤ 46.}, added-at = {2016-04-17T13:19:11.000+0200}, author = {Schmidt, Bernhard}, biburl = {https://www.bibsonomy.org/bibtex/27d09930e143b99414539a5f8183bf9d7/ytyoun}, doi = {10.1023/A:1008398319853}, interhash = {5c277cf35399d32a19139594e3e83397}, intrahash = {7d09930e143b99414539a5f8183bf9d7}, issn = {1573-7586}, journal = {Designs, Codes and Cryptography}, keywords = {hadamard williamson}, number = 1, pages = {61--68}, timestamp = {2016-04-17T13:19:30.000+0200}, title = {{Williamson} Matrices and a Conjecture of {Ito's}}, volume = 17, year = 1999 }