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Relative difference sets with n = 2

โœ Scribed by K.T. Arasu; Dieter Jungnickel; Siu Lun Ma; Alexander Pott

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
866 KB
Volume
147
Category
Article
ISSN
0012-365X

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โœฆ Synopsis

We investigate the existence of relative (m, 2, k, 2)-difference sets in a group H x N relative to N. One can think of these as 'liftings' or 'extensions' of (m, k, 22)-difference sets. We have to distinguish between the difference sets and their complements.. In particular, we prove:

--Difference sets with the parameters of the classical Singer difference sets describing PG(d, q) never admit liftings to relative difference sets with n = 2.

--Difference sets of McFarland and Spence type cannot be extended to relative difference sets with n = 2 (with possibly a few exceptions).

--Paley difference sets are not liftable.

--Twin prime power difference sets and their complements never lift.

--Menon-Hadamard difference sets cannot be extended to relative difference set with n = 2 if the order of the difference set is not a solution of a certain PeUian equation.

Our results give strong evidence for the following conjecture: The only non-trivial difference sets which admit extensions to relative difference sets with n = 2 have the parameters of the complements of Singer difference sets with even dimension.

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