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Probl. Peredachi Inf., 2011, Volume 47, Issue 2, Pages 52–71 (Mi ppi2045)  

This article is cited in 5 scientific papers (total in 5 papers)

Coding Theory

Steiner systems $S(v,k,k-1)$: components and rank

V. A. Zinoviev, D. V. Zinoviev

Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow

Abstract: For an arbitrary Steiner system $S(v,k,t)$, we introduce the concept of a component as a subset of a system which can be transformed (changed by another subset) without losing the property for the resulting system to be a Steiner system $S(v,k,t)$. Thus, a component allows one to build new Steiner systems with the same parameters as an initial system. For an arbitrary Steiner system $S(v,k,k-1)$, we provide two recursive construction methods for infinite families of components (for both a fixed and growing k). Examples of such components are considered for Steiner triple systems $S(v,3,2)$ and Steiner quadruple systems $S(v,4,3)$. For such systems and for a special type of so-called normal components, we find a necessary and sufficient condition for the 2-rank of a system (i.e., its rank over $\mathbb F_2$) to grow under switching of a component. It is proved that for $k\ge5$ arbitrary Steiner systems $S(v,k,k-1)$ and $S(v,k,k-2)$ have maximum possible 2-ranks.

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English version:
Problems of Information Transmission, 2011, 47:2, 130–148

Bibliographic databases:

UDC: 621.391.1+519.7
Received: 20.10.2009
Revised: 18.01.2011

Citation: V. A. Zinoviev, D. V. Zinoviev, “Steiner systems $S(v,k,k-1)$: components and rank”, Probl. Peredachi Inf., 47:2 (2011), 52–71; Problems Inform. Transmission, 47:2 (2011), 130–148

Citation in format AMSBIB
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\by V.~A.~Zinoviev, D.~V.~Zinoviev
\paper Steiner systems $S(v,k,k-1)$: components and rank
\jour Probl. Peredachi Inf.
\yr 2011
\vol 47
\issue 2
\pages 52--71
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\jour Problems Inform. Transmission
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\vol 47
\issue 2
\pages 130--148
\crossref{https://doi.org/10.1134/S0032946011020050}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. D. I. Kovalevskaya, F. I. Solov'eva, “Steiner quadruple systems of small rank embedded into extended perfect binary codes”, J. Appl. Industr. Math., 7:1 (2013), 68–77  mathnet  crossref  mathscinet
    2. D. I. Kovalevskaya, F. I. Solov'eva, E. S. Filimonova, “Steiner triple systems of small rank embedded into perfect binary codes”, J. Appl. Industr. Math., 7:3 (2013), 380–395  mathnet  crossref  mathscinet
    3. D. I. Kovalevskaya, F. I. Solov'eva, “Steiner quadruple systems of small ranks and extended perfect binary codes”, J. Appl. Industr. Math., 7:4 (2013), 522–536  mathnet  crossref  mathscinet
    4. Yu. V. Tarannikov, “O rangakh podmnozhestv prostranstva dvoichnykh vektorov, dopuskayuschikh vstraivanie sistemy Shteinera $S(2,4,v)$”, PDM, 2014, no. 1(23), 73–76  mathnet
    5. M. E. Kovalenko, T. A. Urbanovich, “On the rank of incidence matrices for points and lines of finite affine and projective geometries over a field of four elements”, Problems Inform. Transmission, 50:1 (2014), 79–89  mathnet  crossref  isi
  • Проблемы передачи информации Problems of Information Transmission
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