Williamson Matrices and a Conjecture of Ito's |
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Authors: | Bernhard Schmidt |
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Institution: | (1) Department of Mathematics, MC 253-37, Caltech, Pasadena, CA 91125, USA |
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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 = a, b|a
4t = b
4 = 1, a
2t = b
2, b
-1ab = a-1 for all t of the form t = 2a · 10
b
· 26
c
· m with a, b, c 0, m 1\ (mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over 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 46. |
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Keywords: | Hadamard matrices relative difference sets Williamson matrices Ito's conjecture dicyclic groups |
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