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Directed t-Packings and Directed t-Steiner Systems

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Abstract

The aim of this paper is an investigation of directed t-packings and in particular of directed t-Steiner systems. A new upper bound on the number of points k for directed t-Steiner systems T(t,k,k) is obtained. We disprove a conjecture of Levenshtein on T(t,k,k) for t ≥ 3 by showing that a T(4,6,6) exists. Furthermore, it is proved that the symmetric group S 6 can be partitioned into 30 disjoint T(4,6,6)s. Extensive computer search shows that the tight upper bound on K for t =4,5 is 6 and for t=6 is 7. The non-existence of further small directed t-Steiner systems is established, and large directed t-packings for t,4,5,6 are constructed.

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Authors and Affiliations

  1. Department of Computer Science, University of Toronto, Toronto, Ontario, Canada, M5S3G4

    Rudolf Mathon

  2. Institute for Experimental Mathematics, University of Essen, Ellernstrasse 29, 45326, Essen, Germany

    Tran van Trung

Authors
  1. Rudolf Mathon
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  2. Tran van Trung
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Mathon, R., Trung, T.v. Directed t-Packings and Directed t-Steiner Systems. Designs, Codes and Cryptography 18, 187–198 (1999). https://doi.org/10.1023/A:1008353723204

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