Linked systems of symmetric designs

TL;DR

This paper explores the structure of linked systems of symmetric designs (LSSDs), establishing geometric connections, constructing new examples with large parameters, and analyzing their properties and limitations.

Contribution

It introduces a geometric link between LSSDs and linked simplices, constructs new LSSDs with large parts, and relates LSSDs to equiangular lines and Hadamard matrices.

Findings

Established a connection between LSSDs and linked simplices.

Constructed new LSSDs with arbitrarily large number of parts for fixed parameters.

Showed that most symmetric designs have only two parts based on number theoretic conditions.

Abstract

A linked system of symmetric designs (LSSD) is a $w$-partite graph ($w\geq 2$) where the incidence between any two parts corresponds to a symmetric design and the designs arising from three parts are related. The original construction for LSSDs by Goethals used Kerdock sets, in which $v$ is a power of two. Some four decades later, new examples were given by Davis et.\ al.\ and Jedwab et.\ al.\ using difference sets, again with $v$ a power of two. In this paper we develop a connection between LSSDs and "linked simplices", full-dimensional regular simplices with two possible inner products between vertices of distinct simplices. We then use this geometric connection to construct sets of equiangular lines and to find an equivalence between regular unbiased Hadamard matrices and certain LSSDs with Menon parameters. We then construct examples of non-trivial LSSDs in which $w$ can be made arbitarily large for fixed even part of $v$. Finally we survey the known infinite families of symmetric designs and show, using basic number theoretic conditions, that $w=2$ in most cases.