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

Abstract

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 $$3^n$$ points having 3-rank at most $$3^n-n$$ are resolvable. Combining this observation with the lower bound on the number of such $${\mathrm {STS}}(3^n)$$ recently established by two of the present authors, we obtain a strong lower bound on the number of Kirkman triple systems on $$3^n$$ points. For instance, there are more than $$10^{99}$$ isomorphism classes of $${\mathrm {KTS}}(81)$$ .