An algorithm for enumerating difference sets

TL;DR

This paper introduces an algorithm implemented in the DifSets package for GAP that efficiently enumerates all difference sets in a group up to equivalence, enabling the analysis of larger groups.

Contribution

The paper presents a novel algorithm that reduces the search space for difference sets by leveraging difference sums and quotient groups, allowing enumeration in larger groups.

Findings

Successfully enumerated difference sets in groups of order 64 and 96

Provides a library of known difference sets for research use

Demonstrates the efficiency of the algorithm in reducing computational complexity

Abstract

The DifSets package for GAP implements an algorithm for enumerating all difference sets in a group up to equivalence and provides access to a library of results. The algorithm functions by finding difference sums, which are potential images of difference sets in quotient groups of the original group, and searching their preimages. In this way, the search space can be dramatically decreased, and searches of groups of relatively large order (such as order 64 or order 96) can be completed.