On Resolvability of Steiner Systems $S(v=2^m,4,3)$ of Rank $r\le v-m+1$ over $\mathbb F_2$

Two new constructions of Steiner quadruple systems $S(v,4,3)$ are given. Both preserve resolvability of the original Steiner system and make it possible to control the rank of the resulting system. It is proved that any Steiner system $S(v=2^m,4,3)$ of rank $r\le v-m+1$ over $\mathbb F_2$ is resolvable and that all systems of this rank can be constructed in this way. Thus, we find the number of all different Steiner systems of rank $r=v-m+1$.