Relative difference sets with n = 2

Abstract

We investigate the existence of relative ( m, 2, k, λ)-difference sets in a group H × N relative to N. One can think of these as ‘liftings’ or ‘extensions’ of ( m, k, 2 λ)-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 Pellian 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.