Non-full-rank Steiner quadruple systems $S(v,4,3)$

All different Steiner systems $S(2^m,4,3)$ of order $2^m$ and rank $2^m-m-1+s$ over $\mathbb F_2$, where $0\le s\le m-1$, are constructed. The number of different systems $S(2^m,4,3)$ whose incident matrices are orthogonal to a fixed code is obtained. A connection between the number of different Steiner systems and of different Latin cubes is described.