Relative Difference Sets With N = 2
Document Type
Article
Publication Date
12-16-1995
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, 2λ)-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.
Repository Citation
Arasu, K. T.,
Jungnickel, D.,
Siu Lun Ma, L. M.,
& Pott, A.
(1995). Relative Difference Sets With N = 2. Discrete Mathematics, 147 (1-3), 1-17.
https://corescholar.libraries.wright.edu/math/574
DOI
10.1016/0012-365X(94)00226-9
