Cyclic relative difference sets and circulant weighing matrices

TL;DR

This paper extends the search for cyclic relative difference sets beyond known classical cases and explores their application in constructing circulant weighing matrices.

Contribution

It broadens the classification of cyclic relative difference sets and investigates their potential to generate circulant weighing matrices.

Findings

Extended the search for cyclic relative difference sets beyond classical parameters.

Applied the results to determine the existence of circulant weighing matrices.

Provided new insights into the structure and limitations of relative difference sets.

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.