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Nested Optimal λ-Packings and λ-Coverings of Pairs with Triples

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Abstract

We say that an optimal λ-packing ( λ-covering) of pairs with triples can be nested if it is possible to add one point to each block such that an optimal 2 λ-packing (2 λ-covering) of pairs with quadruples is obtained. A packing (covering) of order v can exist only if v ≥ 4 and v / 3 λ(v - 1) / 2 = v / 4 2λ(v - 1) / 3 ( v / 3 λ(v - 1) / 2 = v / 4 2λ (v - 1) / 3 ). It is shown in this paper that this condition is also sufficient, except possibly for v = 20.

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References

  1. T. Beth, D. Jungnickel and H. Lenz, Design Theory, Bibliographisches Institut, Zurich (1985).

  2. C. J. Colbourn and M. J. Colbourn, Nested triple systems, Ars. Combin., Vol. 16 (1983) pp. 27-34.

    Google Scholar 

  3. W. T. Federer, Construction of classes of experimental designs using transversals in latin squares and Hedayats' sum composition method, Statistical Paper in Honor of George W. Snedecor, Iowa State University Press, Ames, Iowa (1972) pp. 99-114.

    Google Scholar 

  4. H. Hanani, Balanced incomplete block designs and related designs, Discrete Math., Vol. 11 (1975) pp. 255-369.

    Google Scholar 

  5. S. Kageyama and Y. Miao, Existence of nested designs with block size five, J. Statist. Plann. Inference, Vol. 64 (1997) pp. 125-139.

    Google Scholar 

  6. S. Kageyama and Y. Miao, On nested group divisible designs with block size three or four, J. Indian Soc. Agricultural Statist., Vol. 49 (1996/97) pp. 99-110.

    Google Scholar 

  7. J. Q. Longyear, Nested group divisible designs and small nested designs, J. Statist. Plann. Inference, Vol. 13 (1986) pp. 81-87.

    Google Scholar 

  8. D. A. Preece, Nested balanced incomplete block designs, Biometrica, Vol. 54 (1967) pp. 479-486.

    Google Scholar 

  9. J. Schonheim, On coverings, Pacific J. Math., Vol. 14 (1964) pp. 1405-1411.

    Google Scholar 

  10. J. Schonheim, On maximal systems of k-tuples, Studia Sci. Math. Hungar., Vol. 1 (1966) pp. 363-368.

    Google Scholar 

  11. D. R. Stinson, The spectrum of nested Steiner triple systems, Graphs and Combinatorics, Vol. 1 (1985) pp. 189-191.

    Google Scholar 

  12. J. Yin, Some combinatorial constructions for optical orthogonal codes, Discrete Math., Vol. 185 (1998) pp. 245-260.

    Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NF, Canada, A1C 557

    Nabil Shalaby

  2. Department of Mathematics, Suzhou University, Suzhou, 215006, P. R. China

    Jianxing Yin

Authors
  1. Nabil Shalaby
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  2. Jianxing Yin
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Shalaby, N., Yin, J. Nested Optimal λ-Packings and λ-Coverings of Pairs with Triples. Designs, Codes and Cryptography 15, 271–278 (1998). https://doi.org/10.1023/A:1008373226759

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