Steiner triple systems of order 21 with subsystems

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.