Steiner systems S(v, k, k − 1): Components and rank

Steiner systems S(v, k, k − 1): Components and rank 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 ≥ 5 arbitrary Steiner systems S(v, k, k − 1) and S(v, k, k − 2) have maximum possible 2-ranks. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

Steiner systems S(v, k, k − 1): Components and rank

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Publisher
Springer Journals
Copyright
Copyright © 2011 by Pleiades Publishing, Ltd.
Subject
Engineering; Communications Engineering, Networks; Systems Theory, Control; Information Storage and Retrieval; Electrical Engineering
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946011020050
Publisher site
See Article on Publisher Site

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 ≥ 5 arbitrary Steiner systems S(v, k, k − 1) and S(v, k, k − 2) have maximum possible 2-ranks.

Journal

Problems of Information TransmissionSpringer Journals

Published: Jul 14, 2011

References

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