Volume 20, No 3, 2013, P. 3-25

UDC 621.391.15
Kovalevskaya D. I., Solov’eva F. I., Filimonova Е. S. 
Steiner triple systems of small rank embedded into perfect binary codes

Abstract:
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.

Keywords: Steiner triple system, perfect binary code, switching, Pasch configuration, $ijk$-component, $i$-component.

Kovalevskaya Darya Igorevna ¹
Solov’eva Faina Ivanovna ^1,2
Filimonova Elena Sergeevna ¹
1. S. L. Sobolev Institute of Mathematics, SB RAS,
4 Acad. Koptyug Ave., 630090 Novosibirsk, Russia
2. Novosibirsk State University,
2 Pirogov St., 630090 Novosibirsk, Russia
e-mail: daryik@rambler.ru, sol@math.nsc.ru, filimones@rambler.ru