A Mass Formula for Steiner Triple Systems STS(2n−1) of 2-Rank 2n−n

A formula is found for the total number of distinct Steiner triple systems on 2n−1 points whose 2-rank is one higher than the possible minimum 2n−n−1. The formula can be used for deriving bounds on the number of pairwise nonisomorphic systems for large n, and for the classification of all nonisomorphic systems of small orders. It is proved that the number of nonisomorphic Steiner triple systems on 2n−1 points of 2-rank 2n−n grows exponentially.