Symmetric designs and geometroids

Abstract

Aλ-setS in a symmetric 2-(v, k, λ) designΠ is a subset which every block meets in 0, 1 orλ points such that for any point ofS there is a unique block meetingS at that point only. Ovoids in three-dimensional projective spaces are examples ofλ-secs. It is shown that ifπ has aλ-set thenπ is a geometroid withv=λu 2+u+1 andk=λu+1, whereu≧λ−1. The cases whenu isλ−1,λ andλ+1 are investigated and some open problems discussed.

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References

  1. [1]

    E. F. Assmus andJ. H. van Lint, Ovals in projective designs,J. Comb. Theory (A) 27, 307–324 (1979).

    Article  Google Scholar 

  2. [2]

    R. Calderbank andW. M. Kantor, The geometry of two-weight codes,Bull. London Math. Soc. 18, 97–122 (1986).

    Google Scholar 

  3. [3]

    P. Dembrowski, Finite Geometries,New York:Springer (1968).

    Google Scholar 

  4. [4]

    G. L. Ebert, Partitioning projective geometries into caps,Can. J. Math. 37, 6, 1163–1175 (1985).

    Google Scholar 

  5. [5]

    D. R. Hughes, andF. C. Piper: Design Theory.Cambridge University Press, Cambridge (1985).

    Google Scholar 

  6. [6]

    R. C. Mullin, Resolvable designs and geometroids,Utilitas Mathematica 5, 137–149 (1974).

    Google Scholar 

  7. [7]

    D. P. Rajkundlia, Some techniques for constructing infinite families of BIBDs.Discrete Mathematics 44, 61–96 (1983).

    Article  Google Scholar 

  8. [8]

    S. S. Sane, S. S. Shrikhande andN. M. Singhi, Maximal arcs in designs,Graphs and Combinatorics 1, 97–106 (1985).

    Google Scholar 

  9. [9]

    S. S. Shrikhande andN. M. Singhi, Construction of geometroids,Utilitas Mathematica 8, 187–192 (1975).

    Google Scholar 

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McDonough, T.P., Mavron, V.C. Symmetric designs and geometroids. Combinatorica 9, 51–57 (1989). https://doi.org/10.1007/BF02122683

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AMS Subject Classification (1985)

  • 51 E05
  • 05 B05
  • 05B25
  • 51 E20