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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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References

  1. F. E. Bennett and A. Mahmoodi, Directed Designs, The CRC Handbook of Combinatorial Designs (C. J. Colbourn and J. H. Dinitz, eds.) (1996) pp. 317-321.

  2. F. E. Bennett, R. Wei, J. Yin and A. Mahmoodi, Existence of DBIBDs with block size six, Utilitas Math., Vol. 43 (1993) pp. 205-217.

    Google Scholar 

  3. P. A. H. Bours, Codes for Correcting Insertion and Deletion Errors, PhD Thesis, Mathematics Department, Eindhoven Unversity of Technology, 1994.

  4. C. J. Colbourn and A. Rosa, Directed and Mendelsohn triple systems, Contemporary Design Theory: A Collection of Surveys (J. H. Dinitz, D. R. Stinson, eds.), John Wiley & Sons, New York (1992) pp. 97-136.

    Google Scholar 

  5. J. E. Dawson, Jennifer Seberry and D. B. Skillicorn, The Directed Packing Numbers D D(t, v, v), t ≥ 4, Combinatorica, Vol. 4 (1984) pp. 121-130.

    Google Scholar 

  6. H. Hanani, On quadruple systems, Canad. J. Math., Vol. 12 (1960) pp. 145-157.

    Google Scholar 

  7. V. I. Levenshtein, Perfect Deletion-Correcting Codes as Combinatorial Designs, II International Workshop Algebraic and Combinatorial Coding Theory, 16–22 September, Leningrad, USSR, (1990) pp. 137-140.

  8. V. I. Levenshtein, On Perfect Codes in Deletion/Insertion Metric (in Russian), Discretnaya. Mathematika, Vol. 3,No. 1 (1991) pp. 3-20; (in English) Discrete Math. Appl., Vol. 2, No. 3 (1992) pp. 241–258.

    Google Scholar 

  9. A. Mahmoodi, Combinatorial and algorithmic aspects of directed designs, PhD Thesis, University of Toronto (1996).

  10. R. Mathon, Searching for spreads and packings, Geometry, Combinatorial Designs and Related Structures, Proceedings of the First Pythagorean Conference, (J. W. P. Hirschfeld, S. S. Magliveras, M. J. de Resmini, eds.), Cambridge University Press (1997) pp. 161-176.

  11. W. H. Mills, On the covering of triples by quadruples, Proc. 5th S. E. Conf. on Combin., Graph Th. and Comp., Boca Raton, Utilitas, Winnipeg (1974) pp. 573-581.

  12. W. H. Mills and R. C. Mullin, Coverings and packings, Contemporary Design Theory (J. H. Dinitz and D. R. Stinson, eds.), (1992) pp. 371-399.

  13. D. B. Skillicorn and R. G. Stanton, The directed packing numbers D D(t, v, v), Congr. Numer., Vol. 34 (1982) pp. 425-428.

  14. V. D. Tonchev, A class of Steiner 4-wise balanced designs derived from Preparata codes, J. Comb. Designs, Vol. 4 (1996) pp. 203-204.

    Google Scholar 

  15. Tran van Trung, On Ordered Steiner Systems, (Abstract) Thirteenth British Combinatorial Conference, University of Surrey, Guildford, July 8–12 (1991).

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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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