Geodesic
Steiner triple systems of small rank embedded into perfect binary codes
Diskretnyj analiz i issledovanie operacij, Tome 20 (2013) no. 3, pp. 3-25

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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, $ijk$-component, $i$-component.
Mots-clés : Pasch configuration 
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D. I. Kovalevskaya; F. I. Solov'eva; E. S. Filimonova. Steiner triple systems of small rank embedded into perfect binary codes. Diskretnyj analiz i issledovanie operacij, Tome 20 (2013) no. 3, pp. 3-25. http://geodesic.mathdoc.fr/item/DA_2013_20_3_a0/