Cyclic relative difference sets and circulant weighing matrices

Abstract

An $(m,n,k,\lambda)$-relative difference set is a lifting of a $(m,k,n\lambda)$-difference set. Lam gave a table of cyclic relative difference sets with $k \leq 50$ in 1977, all of which were liftings of $( \frac{q^d-1}{q-1},q^{d-1},q^{d-2}(q-1))$-difference sets, the parameters of complements of classical Singer difference sets. Pott found all cyclic liftings of these difference sets with $n$ odd and $k \leq 64$ in 1995. No other nontrivial difference sets are known with liftings to relative difference sets, and Pott ended his survey on relative difference sets asking whether there are any others. In this paper we extend these searches, and apply the results to the existence of circulant weighing matrices.