Skip to main content
SpringerLink
Log in

Existence results for generalized balanced tournament designs with block size 3

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

A generalized balanced tournament design,GBTD(n, k) defined on akn-setV, is an arrangement of the blocks of a (kn, k, k−1)-BIBD defined onV into ann× (kn−1), array such that (1) every element ofV is contained in precisely one cell of each column, and (2) every element ofV is contained in at mostk cells of each row. In this paper, we completely determine the spectrum ofGBTD(n, 3). In addition we prove the exitence of factoredGBTD(n, 3) forn a positive integer,n≥4, with at most one possible exception.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.J.R. Abel,Four mutually orthogonal Latin squares of orders 28 and 52, J. Combinatorial Theory (A), to appear.

  2. A. E. Brouwer,The number of mutually orthogonal Latin squares, Math. Centre report ZW 123, June 1979.

  3. A.E. Brouwer and G.H.J. van Rees,More mutually orthogonal Latin squares, Discrete Mathematics 39 (1982) 263–281.

    Google Scholar 

  4. E.R. Lamken,A note on partitioned balanced tournament designs, Ars Combinatoria 24 (1987) 5–16.

    Google Scholar 

  5. E.R. Lamken,Generalized balanced tournament designs, Transactions of the American Mathematical Society 318, (1990) 473–490.

    Google Scholar 

  6. E.R. Lamken,Constructions for generalized balanced tournament designs, Discrete Mathematics, to appear.

  7. E.R. Lamken,Constructions for resolvable and near resolvable (v, k, k−1)-BIBDs, Coding Theory and Design Theory, Part II, Design Theory, ed. D. Ray-Chaudhuri, Springer-Verlag, IMA Volume 21 (1990) 236–250.

  8. E.R. Lamken,3-complementary frames and doubly near resolvable (v, 3, 2)-BIBDs, Discrete Mathematics 88 (1991) 59–78.

    Google Scholar 

  9. E.R. Lamken,The existence of 3 orthogonal partitioned incomplete Latin squares of type t n, Discrete Mathematics, 89 (1991) 231–251.

    Google Scholar 

  10. E.R. Lamken,On near generalized balanced tournament designs, Discrete Mathematics, 97 (1991) 279–294.

    Google Scholar 

  11. E.R. Lamken,A few more partitioned balanced tournament designs, preprint.

  12. E.R. Lamken,The existence of doubly resolvable (v, 3, 2)-BIBDs, in preparation.

  13. E.R. Lamken,The existence of partitioned generalized balanced tournament designs with block size 3, in preparation.

  14. E.R. Lamken and S.A. Vanstone,The existence of factored balanced tournament designs, Ars Combinatoria 19(1985) 157–160.

    Google Scholar 

  15. E.R. Lamken and S.A. Vanstone,The existence of partitioned balanced tournament designs of side 4n +1, Ars Combinatoria 20 (1985) 29–44.

    Google Scholar 

  16. E.R. Lamken and S.A. Vanstone,The existence of partitioned balanced tournament designs of side 4n +3, Annals of Discrete Mathematics 34 (1987) 319–338.

    Google Scholar 

  17. E.R. Lamken and S.A. Vanstone,The existence of partitioned balanced tournament designs, Annals of Discrete Mathematics 34 (1987) 339–352.

    Google Scholar 

  18. E.R. Lamken and S.A. Vanstone,Balanced tournament designs and related topics, Discrete Mathematics 77 (1989) 159–176.

    Google Scholar 

  19. E.R. Lamken and S.A. Vanstone,Skew transversals in frames, J. of Combinatorial Math and Combinatorial Computing 2 (1987) 37–50.

    Google Scholar 

  20. E.R. Lamken and S.A. Vanstone,Existence results for doubly near resolvable (v, 3, 2)-BIBDs, Discrete Mathematics, to appear.

  21. R. Mathon and A. Rosa,Tables of parameters of BIBDs with r ≤ 41 including existence, enumeration and resolvability results, Annals of Discrete Math 26 (1985) 275–308.

    Google Scholar 

  22. R.C. Mullin,Finite bases for some PBD-closed sets, Discrete Mathematics 77 (1989) 217–236.

    Google Scholar 

  23. E.T. Parker,Nonextendability conditions on mutually orthogonal Latin squares, Proceedings of the American Mathematical Society 13 (1962) 219–221.

    Google Scholar 

  24. R. Roth and M. Peters,Four pairwise orthogonal Latin squares of order 24, J. Combinatorial Theory (A) 44 (1987) 152–155.

    Google Scholar 

  25. P.J. Schellenberg, G.H.J. van Rees and S.A. Vanstone,The existence of balanced tournament designs, Ars Combinatoria 3 (1977) 303–318.

    Google Scholar 

  26. D.R. Stinson,The equivalence of certain incomplete transversal designs and frames, Ars Combinatoria 22 (1986) 81–87.

    Google Scholar 

  27. D.R. Stinson and L. Zhu,On sets of three MOLS with holes, Discrete Mathematics 54 (1985) 321–328.

    Google Scholar 

  28. D.T. Todorov,Three mutually orthogonal Latin squares of order 14 Ars Combinatoria 20 (1985) 45–48.

    Google Scholar 

  29. D.T. Todorov,Three mutually orthogonal Latin squares of order 20, Ars Combinatoria 27 (1989) 63–65.

    Google Scholar 

  30. S.A. Vanstone,Doubly resolvable designs, Discrete Mathematics 29 (1980) 77–86.

    Google Scholar 

  31. S.A. Vanstone,On mutually orthogonal resolutions and near resolutions, Annals of Discrete Mathematics 15 (1982) 357–369.

    Google Scholar 

  32. S.A. Vanstone,private communication.

  33. R.M. Wilson,Concerning the number of mutually orthogonal Latin squares, Discrete Mathematics, (1974) 181–198.

  34. L. Zhu,Pariwise orthogonal Latin squares with orthogonal small subsquares, Research report CORR 83-19, University of Waterloo, Waterloo, Ontario, Canada (1983).

    Google Scholar 

  35. L. Zhu,Incomplete transversal designs with block size five, Congressus Numerantium 69 (1989) 13–20.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Princeton University, 08544, Princeton, NJ

    E. R. Lamken

Authors
  1. E. R. Lamken
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by S.A. Vanstone

This research was supported by a Postdoctoral membership at the Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lamken, E.R. Existence results for generalized balanced tournament designs with block size 3. Des Codes Crypt 3, 33–61 (1993). https://doi.org/10.1007/BF01389354

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01389354

Keywords

Search

Navigation