Relative Difference Sets
A relative $(m,n,k,\lambda)$-difference set in a group $G$ of order $mn$ relative to a normal subgroup $N$ of order $n$ is a subset {$ r_1, r_2, …, r_k $} of $G$ such that the element $R = \Sigma r_i$ in the group ring $Z[G]$ satifies the difference set equation:
$R R^{-1} = n + \lambda (G-N)$,
where $n = k-\lambda$. In other words, the differences of elements in $R$ hit all elements of $G$ exactly $\lambda$ times, except for the “forbidden subgroup” $N$.
It is known that an $(m,n,k,\lambda)$-relative difference set is a lifting of a $(m,k,\lambda n)$-difference set, and a longstanding open problem is: what difference sets lift to relative difference sets? It is conjectured that the only nontrivial cyclic difference sets that lift to relative difference sets have parameters:
\[\left( \frac{q^d-1}{q-1},q^{d-1},q^{d-2}(q-1) \right),\]those of the complements of Singer difference sets.
My forthcoming paper in Journal of Algebraic Combinatorics gives results on a search for lifts of difference sets, showing that up to $k=256$ these are the only known cyclic difference sets with lifts, with four possible exceptions.
Information from this search is at:
