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

The structure of all different Steiner triple systems $S(2^m-1,3,2)$ of rank $2^m-m+2$ over $\mathbb F_2$ is described. This induces a natural recurrent method for constructing Steiner triple systems of any rank. In particular, the method gives all different such systems of order $2^m-1$ and rank $\le2^m-m+2$. The number of such different systems of order $2^m-1$ and rank less than or equal to $2^m-m+2$ which are orthogonal to a given code is found. It is shown that all different triple Steiner systems of order $2^m-1$ and rank $\le2^m-m+2$ are derivative and Hamming. Furthermore, all such triples are embedded in quadruple systems of the same rank and in perfect binary nonlinear codes of the same rank.