Relative (pa, pb, pa, pa−b-Difference Sets: A Unified Exponent Bound and a Local Ring Construction

We show that for an odd prime p the exponent of an abelian group of order pa+b containing a relative (pa, pb, pa, pa−b)-difference set cannot exceed p a/2 +1. Furthermore, we give a new local ring construction of relative (q2u, q, q2u, q2u−1)-difference sets for prime powers q. Finally, we discuss an important open case concerning the existence of abelian relative (pa, p, pa, pa−1)-difference sets.