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By a famous result of Doyen, Hubaut and Vandensavel \cite{DHV}, the 2-rank ofa Steiner triple system on $2^n-1$ points is at least $2^n -1 -n$, and equalityholds only for the classical point-line design in the projective geometry$PG(n-1,2)$. It follows from results of Assmus \cite{A} that, given any integer$t$ with $1 \leq t \leq n-1$, there is a code $C_{n,t}$ containingrepresentatives of all isomorphism classes of STS$(2^n-1)$ with 2-rank at most$2^n -1 -n + t$. Using a mixture of coding theoretic, geometric, designtheoretic and combinatorial arguments, we prove a general formula for thenumber of distinct STS$(2^n-1)$ with 2-rank at most $2^n -1 -n + t$ containedin this code. This generalizes the only previously known cases, $t=1$, provedby Tonchev \cite{T01} in 2001, $t=2$, proved by V. Zinoviev and D. Zinoviev\cite{ZZ12} in 2012, and $t=3$ (V. Zinoviev and D. Zinoviev \cite{ZZ13},\cite{ZZ13a} (2013), D. Zinoviev \cite{Z16} (2016)), while also unifying andsimplifying the proofs. This enumeration result allows us to prove lower andupper bounds for the number of isomorphism classes of STS$(2^n-1)$ with 2-rankexactly (or at most) $2^n -1 -n + t$. Finally, using our recent systematicstudy of the ternary block codes of Steiner triple systems \cite{JT}, we obtainanalogous results for the ternary case, that is, for STS$(3^n)$ with 3-rank atmost (or exactly) $3^n -1 -n + t$. We note that this work provides the firsttwo infinite families of 2-designs for which one has non-trivial lower andupper bounds for the number of non-isomorphic examples with a prescribed$p$-rank in almost the entire range of possible ranks.

Counting Steiner triple systems with classical parameters and prescribed rank