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Diskretnyi Analiz i Issledovanie Operatsii, 2013, Volume 20, Issue 3, Pages 3–25
(Mi da729)
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This article is cited in 3 scientific papers (total in 3 papers)
Steiner triple systems of small rank embedded into perfect binary codes
D. I. Kovalevskayaa, F. I. Solov'evaab, E. S. Filimonovaa a Sobolev Institute of Mathematics, 4 Acad. Koptyug Ave., 630090 Novosibirsk, Russia
b Novosibirsk State University, 2 Pirogov St., 630090 Novosibirsk, Russia
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.
Received: 02.08.2012 Revised: 20.03.2013
Citation:
D. I. Kovalevskaya, F. I. Solov'eva, E. S. Filimonova, “Steiner triple systems of small rank embedded into perfect binary codes”, Diskretn. Anal. Issled. Oper., 20:3 (2013), 3–25; J. Appl. Industr. Math., 7:3 (2013), 380–395
Linking options:
https://www.mathnet.ru/eng/da729 https://www.mathnet.ru/eng/da/v20/i3/p3
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Abstract page: | 299 | Full-text PDF : | 171 | References: | 40 | First page: | 2 |
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