New Strongly Regular Graphs Found via Local Search for Partial Difference Sets

TL;DR

This paper employs local search algorithms to discover new partial difference sets, leading to the construction of previously unknown strongly regular graphs and confirming existing conjectures in combinatorial design theory.

Contribution

The authors introduce a novel application of local search to find partial difference sets, resulting in new SRGs and expanding the catalog of known combinatorial structures.

Findings

Discovered PDSs with 62 different parameter sets.

First known SRG constructions for parameters (144,52,16,20) and (147,66,25,33).

Confirmed existence of (64,18,2,6) PDSs in 73 groups of order 64.

Abstract

Strongly regular graphs (SRGs) are highly symmetric combinatorial objects, with connections to many areas of mathematics including finite fields, finite geometries, and number theory. One can construct an SRG via the Cayley Graph of a regular partial difference set (PDS). Local search is a common class of search algorithm that iteratively adjusts a state to (locally) minimize an error function. In this work, we use local search to find PDSs. We found PDSs with 62 different parameter values in 1254 nonisomorphic groups of orders at most 147. Many of these PDSs replicate known results. In two cases, (144,52,16,20) and (147,66,25,33), the PDSs found give the first known construction of SRGs with these parameters. In some other cases, the SRG was already known but a PDS in that group was unknown. This work also corroborates the existence of (64,18,2,6) PDSs in precisely 73 groups of order 64.