BIBDs with κ = 6 and λ = 1

  • W. H. Mills


Balanced incomplete block designs, or BIBDs, have been studied for many years. In a famous 115 page paper Haim Hanani completely settled the existence question for BIBDs with block size less than 6. In the same paper he settled this question for block size 6 and λ > 1. However when the block size is 6 and λ = 1 the whole question becomes much more difficult. In the present paper we discuss the work that has been done on this. There are, at present, 55 values of υ for which the existence of such a design is in doubt. We will show how the remaining values of υ are handled.


Automorphism Group Base Block Large Block Prime Power Cube Root 
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  1. [1]
    R. J. R. Abel, Four mutually orthogonal Latin squares of orders 28 and 52, J. Combinatorial Theory 58A (1991), 306–309.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    R. J. R. Abel and W. H. Mills, Some new BIBDS with k = 6 and a = 1, J. Combin. Des. 3 (1995), 381–392.MathSciNetzbMATHGoogle Scholar
  3. [3]
    R. J. R. Abel and D. T. Todorov, Four MOLS of orders 20, 30, 38, and 44, J. Combinatorial Theory 64A (1993), 144–148.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    T. Beth, D. Jungnickel, and H. Lenz, Design Theory, Bibliographisches Institut Mannheim, Wien, Zürich, 1985.Google Scholar
  5. [5]
    R. C. Bose, On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements, Calcutta Math. Soc. Golden Jubilee (1959), 341–354.Google Scholar
  6. [6]
    A. E. Brouwer and G. H. J. van Rees, More mutually orthogonal Latin squares, Discrete Math. 39 (1982), 263–281.CrossRefzbMATHGoogle Scholar
  7. [7]
    R. H. Bruck and H. J. Ryser, The nonexistence of certain finite projective planes, Canad. J. Math. 1 (1949), 88–93.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Charles J. Colbourn, Four MOLS of order 26, J. Combin. Math. Combin. Comput. 17 (1995), 147–148.MathSciNetzbMATHGoogle Scholar
  9. [9]
    R. H. F. Denniston, A Steiner system with a maximal arc, Ars Combinatoria 9 (1980), 247–248.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Malcolm Greig, unpublished.Google Scholar
  11. [11]
    Marshall Hall, Jr., Combinatorial Theory, John Wiley and Sons, New York (1986).Google Scholar
  12. [12]
    Haim Hanani, Balanced incomplete block designs and related designs, Discrete Math. 11 (1975), 255–369.CrossRefzbMATHGoogle Scholar
  13. [13]
    Diane M. Johnson, A. L. Dulmage, and N. S. Mendelsohn, Orthomorphisms of groups and orthogonal Latin squares I, Canad. J. Math. 13 (1961), 356–372.MathSciNetzbMATHGoogle Scholar
  14. [14]
    H. F. MacNeish, Euler squares, Ann. Math. 23 (1922), 221–227.MathSciNetGoogle Scholar
  15. [15]
    R. D. McKay and R. G. Stanton, Isomorphism of two large designs, Ars Combinatoria 6 (1978), 87–90.MathSciNetzbMATHGoogle Scholar
  16. [16]
    Y. Miao and Lie Zhu, On resolvable BIBDs with block size five Ars Corn bin. 24 (1995), 261–275.MathSciNetGoogle Scholar
  17. [17]
    W. H. Mills, Two new block designs, Utilitas Math. 7 (1975), 73–75.zbMATHGoogle Scholar
  18. [18]
    W. H. Mills, A new block design, Congressus Num. 14 (1975), 461–465.Google Scholar
  19. [19]
    W. H. Mills, Some mutually orthogonal Latin squares, Congressus Num. 19 (1977), 473–487.Google Scholar
  20. [20]
    W. H. Mills, The construction of balanced incomplete block designs with A = 1, Congressus Num. 20 (1977), 131–148.Google Scholar
  21. [21]
    W. H. Mills, The construction of BIBDs using nonabelian groups, Con- gressus Num. 21 (1978), 519–526.Google Scholar
  22. [22]
    W. H. Mills, The construction of balanced incomplete block designs, Con- gressus Num. 23 (1979), 73–86.Google Scholar
  23. [23]
    W. H. Mills, Balanced incomplete block designs with k = 6 and A = 1, Enumeration and Design, Academic Press, Toronto, Ont. (1984), 239–244.Google Scholar
  24. [24]
    R. C. Mullin, Finite bases for some PBD-closed sets, Discrete Math. 77 (1989), 217–236.MathSciNetzbMATHGoogle Scholar
  25. [25]
    R. C. Mullin, D. G. Hoffman, and C. C. Lindner, A few more BIBD’s with k = 6 and A = 1, Combinatorial Design Theory, North-Holland Math. Stud. 149, North Holland, Amsterdam-New York, (1987), 379–384.Google Scholar
  26. [26]
    Charles E. Roberts, Jr., Sets of mutually orthogonal Latin squares with `like subsquares’, J. Combin. Theory 61A (1992), 50–63.CrossRefzbMATHGoogle Scholar
  27. [27]
    R. G. Stanton and D. G. Gryte, A family of BIBDs, Combinatorial Structures and their Applications,Gordon and Breach, New York (1970), 411412.Google Scholar
  28. [28]
    R. M. Wilson, Cyclotomy and difference families in elementary abelian groups, J. Number Theory 4 (1972), 17–47.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    R. M. Wilson, An existence theory for pairwise balanced designs, I: Compo- sition theorems and morphisms, J. Combin. Theory 13A (1972), 220–245.CrossRefzbMATHGoogle Scholar
  30. [30]
    R. M. Wilson, Concerning the number of mutually orthogonal Latin squares, Discrete Math. 9 (1974), 181–198.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    R. M. Wilson, An existence theory for pairwise balanced designs, III: Proof of the existence conjectures, J. Combin. Theory 18A (1975), 71–79.CrossRefzbMATHGoogle Scholar
  32. [32]
    Lie Zhu, Beiliang Du, and Yin Jianxing, Some new balanced incomplete block designs with k = 6 and A = 1, Ars Combin. 24 (1987), 167–174.MathSciNetzbMATHGoogle Scholar
  33. [33]
    Lie Zhu, Beiliang Du, and Xuebin Zhang, A few more RBIBDs with k = 5, A = 1, Discrete Math. 97 (1991), 409–417.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • W. H. Mills
    • 1
  1. 1.Institute for Defense AnalysesPrincetonUSA

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