Steiner triple systems of small rank embedded into perfect binary codes

Using the switching method, we classify Steiner triple systems $\mathrm{STS}(n)$ of order $n=2^r-1$, $r>3$, and of small rank $r_n$ (which differs by 2 from the rank of the Hamming code of length $n$) embedded into perfect binary codes of length $n$ and of the same rank. The lower and upper bounds for the number of such different $\mathrm{STS}$ are given. We present the description and the lower bound for the number of $\mathrm{STS}(n)$ of rank $r_n$ which are not embedded into perfect binary codes of length $n$ and of the same rank. The embeddability of any $\mathrm{STS}(n)$ of rank $r_n-1$ into a perfect code of length $n$ with the same rank, given by Vasil’ev construction, is proved. Bibliogr. 22.