Steiner Triple Systems of Order 21 with Subsystems

TL;DR

This paper completes the classification of Steiner triple systems of order 21 by determining the number of systems with sub-STS(7)s, building on prior work that classified those with sub-STS(9)s.

Contribution

It provides the first complete enumeration of STS(21)s with sub-STS(7)s, expanding the understanding of their structure and isomorphism classes.

Findings

116,635,963,205,551 isomorphism classes with sub-STS(7)s

12,661,527,336 isomorphism classes with sub-STS(9)s

Estimation of total isomorphism classes of STS(21)s

Abstract

The smallest open case for classifying Steiner triple systems is order 21. A Steiner triple system of order 21, an STS(21), can have subsystems of orders 7 and 9, and it is known that there are 12,661,527,336 isomorphism classes of STS(21)s with sub-STS(9)s. Here, the classification of STS(21)s with subsystems is completed by settling the case of STS(21)s with sub-STS(7)s. There are 116,635,963,205,551 isomorphism classes of such systems. An estimation of the number of isomorphism classes of STS(21)s is given.