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(22, 33, 12, 8, 4)-BIBD, an Update

  • G. H. J. van Rees

Abstract

A (v, b,r, k, λ)-balanced incomplete block design is a family of b sets (called blocks) of size k whose elements (varieties) are from a v-set, v > k, such that every element occurs exactly r times and every pair exactly λ times. A (22, 33, 12, 8, 4)-BIBD is the set of parameters with the smallest v for which it is not known whether a BIBD exists or not. A survey of what is known about such a design is given.

Keywords

Automorphism Group Element Orbit Incidence Matrix Dual Code Steiner Triple System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    J.A. Bate, M. Hall Jr., and G.H.J. van Rees. Structures within (22,33,12,8,4)-designs. J. Comb. Math. Comb. Comp., 4 (1988), 115–122.MATHGoogle Scholar
  2. [2]
    R.E. Block. On the orbits of collineation groups. Math. Zeitschr., 96 (1967), 33–49.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    C. Demeng, M. Greig, and G.H.J. van Rees. non-existent preprint.Google Scholar
  4. [4]
    R.A. Fisher and F. Yates. Statistical Tables for Biological, Agricultural and Medical Research, volume 93. Hafner, New York, 6th edition edition, 1963.Google Scholar
  5. [5]
    M. Greig. An improvement to connor’s criterion. preprint.Google Scholar
  6. [6]
    M. Hall Jr. Constructive methods for designs. Cong. Numer., 66 (1988), 141–144.Google Scholar
  7. [7]
    M. Hall Jr., R. Roth, G.H.J. van Rees, and S.A. Vanstone. On designs (22,33,12,8,4). J. Combinatorial Theory, 47 (1988), 157–175.CrossRefMATHGoogle Scholar
  8. [8]
    N. Hamada and Y. Kobayshi. On the block structure of BIB designs with parameters v = 22, b = 33, r = 12, k = 8 and λ = 4. J. Combinatorial Theory, 24A (1978), 75–83.CrossRefMATHGoogle Scholar
  9. [9]
    S. Kapralov. Combinatorial 2-(22,8,4) designs with automorphisms of order 3 fixing one point. In Math. and Education in Math., Proc. of the XVI Spring Conference of Union of Bulgarian Mathematicians, pages 453–458. Sunny Beach, 1987.Google Scholar
  10. [10]
    I. Landgev and V. Tonchev. Automorphisms of 2-(22,8,4) designs. Discrete Math., 77 (1989), 177–189.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    F.J. MacWilliams. A theorem on the distribution of weights in a systematic code. Bell System Tech. J., 42 (1963), 79–94.MathSciNetGoogle Scholar
  12. [12]
    R. Mathon and A. Rosa. Tables of parameters of BIBDs with r <= 41 including existence, enumeration and resolvability results: An update. Ars Combin., 30 (1991), 65–96.MathSciNetGoogle Scholar
  13. [13]
    B.D. McKay and S.P. Radziszowski. Towards deciding the existence of 2-(22,8,4) designs. J. Comb. Math. Comb. Comp.. submitted.Google Scholar
  14. [14]
    V. Pless. A classification of self-orthogonal codes over GF(2). Discrete Math., 3 (1972), 209–246.MathSciNetMATHGoogle Scholar
  15. [15]
    V. Pless and N.J.A. Sloane. On the classification and enumeration of self-dual codes. J. Combinatorial Theory, 18A (1975), 313–335.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    J. Siftar. On 2-groups operating on a 2-(22,8,4) design. Rad.-Mat., 7 (2) (1991), 217–229.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • G. H. J. van Rees
    • 1
  1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCanada

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