Taming the Differences: Exploring Paradigm Shifts in the Construction of Difference Sets

In this information-driven world, discrete mathematics forms the backbone of many essential technologies and systems. At the heart of this field are objects with strong regularity, which are central to the branch of mathematics known as combinatorics. The current project studies difference sets and partial difference sets, both of which can be neatly described as subsets exhibiting remarkable regularity. These are the underlying objects behind many elegant configurations in a wide range of areas including, on the theory side: finite geometry, coding theory, combinatorial design theory, number theory, graph theory, and on the application side: sequence design, signal processing, information security. Despite their wide-reaching significance, finding explicit constructions of these sets has remained a long-standing challenge. This project embraces an ongoing paradigm shift in the approach to constructing such objects, with the goal of uncovering novel constructions that will substantially advance the constructive landscape of difference sets and partial difference sets. The PI plans to involve graduate students. This project investigates the central challenge in the study of difference sets and partial difference sets: their explicit constructions. Inspired by the recent breakthrough of Applebaum et al., which determined exactly which of the 56,092 non-isomorphic groups of order 256 contain a difference set, the project will pursue three key directions: (1) constructing difference sets in abelian groups with high exponents, (2) generalizing the classical Denniston partial difference sets to a wider range of parameters and groups, (3) advancing construction techniques beyond abelian settings to the comparatively unexplored territory of nonabelian groups. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.