# The Steiner triple systems of order 21 with a transversal subdesign TD(3,6)

## TL;DR

This paper investigates the structure of Steiner triple systems of order 21 containing transversal subdesigns, identifying over two million isomorphism classes through exhaustive computational analysis.

## Contribution

It provides a comprehensive classification of STS(21) with transversal subdesigns, revealing the number of such systems and their structural properties.

## Key findings

- Over 2 million isomorphism classes of STS(21) with a TD(3,6) subdesign identified
- Structural properties of these systems are characterized
- Connection established between STS(21) and 6x6 Latin squares

## Abstract

We prove several structural properties of Steiner triple systems (STS) of order 3w+3 that include one or more transversal subdesigns TD(3,w). Using an exhaustive search, we find that there are 2004720 isomorphism classes of STS(21) including a subdesign TD(3,6), or, equivalently, a 6-by-6 latin square.