New Partial Geometric Difference Sets and Partial Geometric Difference Families

TL;DR

This paper constructs new families of partial geometric difference sets and families, leading to directed strongly regular graphs with novel parameters, and explores their connections with partially balanced designs and 2-adesigns.

Contribution

It introduces new partial geometric difference sets and families with novel parameters, expanding the known classes and their applications in graph theory and design theory.

Findings

Constructed several new families of partial geometric difference sets.

Generated directed strongly regular graphs with new parameters.

Explored links between partial geometric designs, 2-adesigns, and partially balanced designs.

Abstract

Olmez, in "Symmetric $1\frac{1}{2}$-Designs and $1\frac{1}{2}$-Difference Sets" (2014), introduced the concept of a partial geometric difference set (also referred to as a $1\frac{1}{2}$-design), and showed that partial geometric difference sets give partial geometric designs. Nowak et al., in "Partial Geometric Difference Families" (2014), introduced the concept of a partial difference family, and showed that these also give partial geometric designs. It was shown by Brouwer et al. in "Directed strongly regular graphs from $1\frac{1}{2}$-designs" (2012) that directed strongly regular graphs can be obtained from partial geometric designs. In this correspondence we construct several families of partial geometric difference sets and partial difference families with new parameters, thereby giving directed strongly regular graphs with new parameters. We also discuss some of the links between partially balanced designs, $2$-adesigns (which were recently coined by Cunsheng Ding in "Codes from Difference Sets" (2015)), and partial geometric designs, and make an investigation into when a $2$-adesign is partial geometric.