Linking systems of difference sets

TL;DR

This paper investigates the existence and construction of linking systems of difference sets in various groups, introducing new methods for 2-groups and establishing limitations in non-2-groups.

Contribution

It provides a new construction method for linking systems in 2-groups using group difference matrices and explores the boundaries of such systems in non-2-groups.

Findings

Large linking systems in elementary abelian 2-groups via Kerdock and bent sets

New constructions in 2-groups using group difference matrices

First known linking systems in nonabelian groups

Abstract

A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2-groups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in non-2-groups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2-groups. We give a new construction for linking systems of difference sets in 2-groups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2-groups, new linking systems in other 2-groups for which no system was previously known, and the first known examples in nonabelian groups.