Acta Mathematica Sinica

, Volume 16, Issue 1, pp 103–112

The Existence of BIB Designs



Given any positive integers k≥ 3 and λ, let c(k, λ) denote the smallest integer such that vB(k, λ) for every integer vc(k, λ) that satisfies the congruences λv(v− 1) ≡ 0(mod k(k− 1)) and λ(v− 1) ≡ 0(mod k− 1). In this article we make an improvement on the bound of c(k, λ) provided by Chang in [4] and prove that \( c{\left( {k,\lambda } \right)} \leqslant \exp {\left\{ {k^{{3k^{6} }} } \right\}} \). In particular, \( c{\left( {k,1} \right)} \leqslant \exp {\left\{ {k^{{k^{2} }} } \right\}} \).


Wilson’s theorem Balanced incomplete block design PBD-closed 

1991 MR Subject Classification



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  1. 1.
    R M Wilson. An existence theory for pairwise blanced designs, I, II, III. J Combin Theory (Ser A), 1972, 13: 220-245, 246-273, 1975, 18: 71-79MATHCrossRefGoogle Scholar
  2. 2.
    Yanxun Chang. A bound for Wilson's theorem (I). J Combin Design, 1995, 3: 25-39MATHGoogle Scholar
  3. 3.
    Yanxun Chang. A bound for Wilson's theorem (II). J Combin Design, 1996, 4(1): 11-26MATHCrossRefGoogle Scholar
  4. 4.
    Yanxun Chang. A bound for Wilson's theorem (III). J Combin Design, 1996, 4(2): 83-93MATHCrossRefGoogle Scholar
  5. 5.
    H Hanani. Balanced incomplete block designs and related designs. Discrete Math, 1975, 11: 255-369MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Lie Zhu. Some recent development on BIBD and related designs. Discrete Math, 1993, 123: 189-214MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    R J R Abel, W H Mills. Some new BIBDs with k = 6 and λ = 1. J Combin Design, 1995, 5: 381-391MathSciNetGoogle Scholar
  8. 8.
    R J R Abel, M Greig. Some new RBIBDs with block size 5 and PBDs with block sizes ≡ 1(mod 5). Australian J Combinatorics, 1997, 15: 177-202MATHMathSciNetGoogle Scholar
  9. 9.
    M Greig. Balanced incomplete block designs with a block size of 7. preprintGoogle Scholar
  10. 10.
    R M Wilson. Construction and uses of pairwise balanced designs. Mathematical Centre Tracts, 1974, 55: 18-41Google Scholar
  11. 11.
    Yanxun Chang. On the estimate of the number of mutually orthogonal Latin squares. JCMCC, 1996, 21: 217-222MATHGoogle Scholar
  12. 12.
    Yanxun Chang. On the existence of BIB designs with large λ. to appear in J Statistical Planning and InferenceGoogle Scholar
  13. 13.
    T Beth, D Jungnickel, H Lenz. Design Theory. Cambridge University Press, Cambridge, 1986Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  1. 1.Institute of MathematicsNorthern Jiaotong UniversityBeijing 100044P.R. China

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