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Generalized Steiner Triple Systems with Group Size g ≡ 0,3 (mod 6)

Generalized Steiner Triple Systems with Group Size g ≡ 0,3 (mod 6) Generalized Steiner triple systems, (GS(2,3,n,g), are equivalent to maximum constant weight codes over an alphabet of size g+1 with distance 3 and weight 3 in which each codeword has length n. The necessary conditions for the existence of a GS(2,3,n,g) are (n−1)g≡0 (mod 2), n(n−1)g 2≡0 (mod 6), and n≥g+2. These necessary conditions are shown to be sufficient by several authors for 2≤g≤11. In this paper, three new results are obtained. First, it is shown that for any given g, g≡0 (mod 6) and g≥12, if there exists a GS(2,3,n,g) for all n, g+2≤n≤7g+13, then the necessary conditions are also sufficient. Next, it is also shown that for any given g, g≡3 (mod 6) and g≤15, if there exists a GS(2,3,n,g) for all n, n≡1 (mod 2) and g+2≤n≤7g+6, then the necessary conditions are also sufficient. Finally, as an application, it is proved that the necessary conditions for the existence of a GS(2,3,n,g) are also sufficient for g=12,15. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Generalized Steiner Triple Systems with Group Size g ≡ 0,3 (mod 6)

Acta Mathematicae Applicatae Sinica , Volume 18 (4) – Jan 1, 2002

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Publisher
Springer Journals
Copyright
Copyright © 2002 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s102550200058
Publisher site
See Article on Publisher Site

Abstract

Generalized Steiner triple systems, (GS(2,3,n,g), are equivalent to maximum constant weight codes over an alphabet of size g+1 with distance 3 and weight 3 in which each codeword has length n. The necessary conditions for the existence of a GS(2,3,n,g) are (n−1)g≡0 (mod 2), n(n−1)g 2≡0 (mod 6), and n≥g+2. These necessary conditions are shown to be sufficient by several authors for 2≤g≤11. In this paper, three new results are obtained. First, it is shown that for any given g, g≡0 (mod 6) and g≥12, if there exists a GS(2,3,n,g) for all n, g+2≤n≤7g+13, then the necessary conditions are also sufficient. Next, it is also shown that for any given g, g≡3 (mod 6) and g≤15, if there exists a GS(2,3,n,g) for all n, n≡1 (mod 2) and g+2≤n≤7g+6, then the necessary conditions are also sufficient. Finally, as an application, it is proved that the necessary conditions for the existence of a GS(2,3,n,g) are also sufficient for g=12,15.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2002

References