Steiner triple systems $S(2^m-1,3,2)$ of rank $2^m-m+1$ over $\mathbb F_2$

Steiner systems $S(2^m-1,3,2)$ of rank $2^m-m+1$ over the field $\mathbb F_2$ are considered. A new recursive method for constructing Steiner triple systems of an arbitrary rank is proposed. The number of all Steiner systems of rank $2^m-m+1$ is obtained. Moreover, it is shown that all Steiner triple systems $S(2^m-1,3,2)$ of rank $r\le2^m-m+1$ are derived, i.e., can be completed to Steiner quadruple systems $S(2^m,4,3)$. It is also proved that all such Steiner triple systems are Hamming; i.e., any Steiner triple system $S(2^m-1,3,2)$ of rank $r\le2^m-m+1$ over the field $\mathbb F_2$ occurs as the set of words of weight $3$ of a binary nonlinear perfect code of length $2^m-1$.