Steiner triple systems of order 21 with subsystems

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

Bussey systems and Steiner\u27s tactical problem

In 1853, Steiner posed a number of combinatorial (tactical) problems, which eventually led to a large body of research on Steiner systems. However, solutions to Steiner\u27s questions coincide with Steiner systems only for strengths two and three. For larger strengths, essentially only one class of solutions to Steiner\u27s tactical problems is known, found by Bussey more than a century ago. In this paper, the relationships among Steiner systems, perfect binary one-error-correcting codes, and solutions to Steiner\u27s tactical problem (Bussey systems) are discussed. For the latter, computational results are provided for at most 15 points