Designs, Codes and Cryptography

, Volume 10, Issue 3, pp 275–307 | Cite as

Existence of Incomplete Transversal Designs with Block Size Five and Any Index λ

  • R. J. R. Abel
  • Charles J. Colbourn
  • Jianxing Yin
  • Hantao Zhang


The basic necessary condition for the existence of a TD(5, λ; v)-TD(5, λ; u), namely v ≥ 4u, is shown to be sufficient for any λ ≥ 1, except when (v, u) = (6, 1) and λ = 1, and possibly when (v, u) = (10, 1) or (52, 6) and λ = 1. For the case λ = 1, 86 new incomplete transversal designs are constructed. Several construction techniques are developed, and some new incomplete TDs with block size six and seven are also presented.

transversal design latin square group divisible design pairwise balanced design orthgonal array 


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  1. 1.
    R. J. R. Abel, Four mutually orthogonal latin squares of order 28 and 52, J. Combin. Theory (A), Vol. 58 (1991) pp. 306–309.Google Scholar
  2. 2.
    R. J. R. Abel, On the existence of balanced incomplete block designs and transversal designs, Ph.D. thesis, University of New South Wales (1995).Google Scholar
  3. 3.
    R. J. R. Abel, C. J. Colbourn and J. H. Dinitz, Incomplete MOLS: CRC Handbook of Combinatorial Designs (C. J. Colbourn and J. H. Dinitz, eds.), CRC Press, Boca Raton, FL (1996) pp. 142–172.Google Scholar
  4. 4.
    R. J. R. Abel and D. T. Todorov, Four MOLS of order 20, 30, 38 and 44, J. Combin. Theory (A), Vol. 64 (1993) pp. 144–148.Google Scholar
  5. 5.
    T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge University Press, Cambridge (1986).Google Scholar
  6. 6.
    R. C. Bose, S. S. Shrikhande and E. T. Parker, Further results on the construction of mutually orthogonal latin squares and the falsity of Euler's conjecture, Canad. J. Math., Vol. 12 (1960) pp. 189–203.Google Scholar
  7. 7.
    A. E. Brouwer, Four MOLS of order 10 with a hole of order 2, J. Statist. Planning and Inference, Vol. 10 (1984) pp. 203–205.Google Scholar
  8. 8.
    A. E. Brouwer, The number of mutually orthogonal latin squares—A table up to order 10000, Math. Cent. Report ZW 123/79.Google Scholar
  9. 9.
    A. E. Brouwer and G. H. J. van Rees, More mutually orthogonal latin squares, Discrete Math., Vol. 39 (1982) pp. 263–281.Google Scholar
  10. 10.
    C. J. Colbourn, Four MOLS of order 26, J. Comb. Math. Comb. Comput., Vol. 17 (1995) pp. 147–148.Google Scholar
  11. 11.
    C. J. Colbourn, Some direct constructions for incomplete transversal designs, J. Statist. Planning and Inference, to appear.Google Scholar
  12. 12.
    C. J. Colbourn, Construction techniques for mutually orthogonal latin squares: Combinatorics Advances (C. J. Colbourn and E. S. Mahmoodian, eds.) Kluwer Academic Press (1995) pp. 27–48.Google Scholar
  13. 13.
    C. J. Colbourn, J. H. Dinitz and M. Wojtas, Thwarts in transversal designs, Des. Codes Crypt., Vol. 5 (1995) pp. 189–197.Google Scholar
  14. 14.
    D. A. Drake and H. Lenz, Orthogonal latin squares with orthogonal subsequares, Archiv. der Math., Vol. 34 (1980) pp. 565–576.Google Scholar
  15. 15.
    B. Du, On the existence of incomplete transversal designs with block size five, Discrete Math., Vol. 135 (1994) pp. 81–92.Google Scholar
  16. 16.
    R. Fuji-Hara and S. A. Vanstone, On the spectrum of doubly resolvable Kirkman systems, Congressus Numerantium, Vol. 28 (1980) pp. 399–407.Google Scholar
  17. 17.
    S. Furino, Y. Miao and J. Yin, Frames and Resolvable Designs, CRC Press, Boca Raton, FL, to appear.Google Scholar
  18. 18.
    A. H. Hamel, W. H. Mills, R. C. Mullin, R. Rees, D. R. Stinson and J. Yin, The spectrum of PBD({5, k*}, v) for k = 9, 13, Ars Combinatoria, Vol. 36 (1993) pp. 7–26.Google Scholar
  19. 19.
    H. Hanani, Balanced incomplete block designs and related designs, Discrete Math., Vol. 11 (1975) pp. 255–369.Google Scholar
  20. 20.
    K. Heinrich, Near-orthogonal latin squares, Utilitas Math., Vol. 12 (1977) pp. 145–155.Google Scholar
  21. 21.
    K. Heinrich and L. Zhu, Existence of orthogonal latin squares with aligned subsquares, Discrete Math., Vol. 59 (1986) pp. 241–248.Google Scholar
  22. 22.
    J. D. Horton, Sub-latin squares and incomplete orthogonal arrays, J. Combinatorial Theory (A), Vol. 16 (1974) pp. 23–33.Google Scholar
  23. 23.
    E. R. Lamken, The existence of orthogonal partitioned incomplete latin squares of type t n, Discrete Math., Vol. 89 (1991) pp. 231–251.Google Scholar
  24. 24.
    R. C. Mullin, private communication, 1994.Google Scholar
  25. 25.
    R. C. Mullin, A generalization of the singular direct product with applications to skew Room squares, J. Combinatorial Theory (A), Vol. 29 (1980) pp. 306–318.Google Scholar
  26. 26.
    R. C. Mullin and D. R. Stinson, Pairwise balanced designs with block sizes 6t + 1, Graphs Combin., Vol. 3 (1987) pp. 365–377.Google Scholar
  27. 27.
    R. C. Mullin, P. J. Schellenberg, D. R. Stinson and S. A. Vanstone, Some results on the existence of squares, Ann. Discrete Math., Vol. 6 (1980) pp. 257–274.Google Scholar
  28. 28.
    T. G. Ostrom and F. A. Sherk, Finite projective planes with affine subplanes, Canad. Math. Bull, Vol. 7 (1964) pp. 549–560.Google Scholar
  29. 29.
    E. T. Parker, Nonextendibility conditions on mutually orthogonal latin squares, Proc. Amer. Math. Soc., Vol. 13 (1962) pp. 219–221.Google Scholar
  30. 30.
    J. F. Rigby, Affine subplanes of finite projective planes, Canad. J. Math., Vol. 17 (1965) pp. 977–1009.Google Scholar
  31. 31.
    R. Roth and M. Peters, Four pairwise orthogonal latin squares of order 24, J. Combin. Theory (A), Vol. 44 pp. 152–155.Google Scholar
  32. 32.
    W. D. Wallis and L. Zhu, Orthogonal latin squares with small subsquares: Combinatorial Mathematics, 10(Adelaide, 1982), Lecture Notes in Math., Springer, Berlin, 1036 (1983) pp. 398–409.Google Scholar
  33. 33.
    R. M. Wilson, Concerning the number of mutually orthogonal latin squares, Discrete Math., Vol. 9 (1974) pp. 181–198.Google Scholar
  34. 34.
    R. M. Wilson, A few more squares: Proc. 5th S-E Conf., On Combinatorics, Graph Theory and Computing, (1974) pp. 675–680.Google Scholar
  35. 35.
    R. M. Wilson, Constructions and use of pairwise balanced designs, Math. Centre Tracts, Vol. 55 (1974) pp. 18–41.Google Scholar
  36. 36.
    J. Yin, On the packing of pairs by quintuples with index 2, Ars Combin., Vol. 31 (1991) pp. 287–301.Google Scholar
  37. 37.
    L. Zhu, Some results on orthogonal latin squares with orthogonal subsquares, Utilitas Math., Vol. 25 (1984) pp. 241–248.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • R. J. R. Abel
    • 1
  • Charles J. Colbourn
    • 2
  • Jianxing Yin
    • 3
  • Hantao Zhang
    • 4
  1. 1.School of MathematicsUniversity of New South WalesKensingtonAustralia
  2. 2.Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  3. 3.Department of MathematicsSuzhou UniversitySuzhouPeople's Republic of China
  4. 4.Computer ScienceUniversity of IowaIowa CityU.S.A

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