On ranks of subsets in the space of binary vectors admitting an embedding of a Steiner system $S(2,4,v)$

A bound for the rank of a subset $X$ in the vector space $\mathbb F_2^n$ is obtained via the covering radius of the code lying in the subspace of linear dependencies of vectors in $X$. Also, an upper bound for the covering radius of a code generated by the incidence matrix of a Steiner system $S(2,4,v)$ is obtained. Precice and asymptotic bounds for the rank of a subset $X$ in the vector space $\mathbb F_2^n$ admitting an embedding of a Steiner system $S(2,4,v)$ are obtained too.