On New Families of Supplementary Difference Sets over Rings with Short Orbits

We discuss difference sets (DS) and supplementary difference sets (SDS) over rings. We survey some constructions of SDS over Galois rings where there are no short orbits. From there, we move to constructions involving short orbits, yielding new infinite families
of SDS over \(\text{GF}(p) \times \text{GF}(q)\), \(p\), \(q\) both prime powers.
Many of these families have \(\lambda = 1\). We also present new balanced incomplete block designs and pairwise balanced designs arising from the constructions given here.