The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

TL;DR

This paper constructs highly symmetric Kirkman triple systems for specific orders, revealing new automorphism group structures and introducing novel difference family and matrix concepts.

Contribution

It provides the first explicit constructions of KTSs with automorphism groups acting sharply transitively on all but three points for certain congruence classes.

Findings

Existence of KTS with at least v-3 automorphisms for specific orders.

Introduction of doubly disjoint difference families.

Introduction of splittable difference matrices.

Abstract

Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We fill this gap by proving that whenever $v \equiv 39$ (mod 72), or $v \equiv 4^e48 + 3$ (mod $4^e96$) and $e \geq 0$, there exists a KTS on $v$ points having at least $v-3$ automorphisms. This is only one of the consequences of a careful investigation on the KTSs with an automorphism group $G$ acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of $G$, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.