Large sets of Kirkman triple systems with order $q^n+2$

TL;DR

This paper presents a new recursive construction method for large sets of Kirkman Triple Systems (LKTS) with order $q^n+2$, expanding the known existence range for these combinatorial designs.

Contribution

It introduces a novel recursive construction of LKTS$(q^n+2)$ from LKTS$(q+2)$ for prime powers $q$ of the form $6t+1$, enabling the discovery of previously unknown LKTS.

Findings

Constructed LKTS with orders 171, 345, 363

Extended the known existence range of LKTS

Provided a new recursive method for LKTS construction

Abstract

The existence of Large sets of Kirkman Triple Systems (LKTS) is an old problem in combinatorics. Known results are very limited, and a lot of them are based on the works of Denniston \cite{MR0349416, MR0369086, MR535159, MR539718}. The only known recursive constructions are an tripling construction by Denniston \cite{MR535159}and a product construction by Lei \cite{MR1931492}, both constructs an LKTS($uv$) on the basis of an LKTS($v$). In this paper, we describe an construction of LKTS$(q^n+2)$ from LKTS$(q+2)$, where $q$ is a prime power of the form $6t+1$. We could construct previous unknown LKTS($v$) by this result, the smallest among them have $v=171,345,363$.