The classification of Steiner triple systems on 27 points with 3-rank 24

We show that there are exactly 2624 isomorphism classes of Steiner triple systems on 27 points having 3-rank 24, all of which are actually resolvable. More generally, all Steiner triple systems on 3n points having 3-rank at most 3n−n are resolvable. Combining this observation with the lower bound on the number of such STS(3n) recently established by two of the present authors, we obtain a strong lower bound on the number of Kirkman triple systems on 3n points. For instance, there are more than 1099 isomorphism classes of KTS(81)