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Problemy Peredachi Informatsii, 2011, Volume 47, Issue 2, Pages 52–71
(Mi ppi2045)
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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.
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
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https://www.mathnet.ru/eng/ppi2045 https://www.mathnet.ru/eng/ppi/v47/i2/p52
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Abstract page: | 550 | Full-text PDF : | 93 | References: | 59 | First page: | 11 |
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