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Quasi-symmetric designs with the difference of block intersection numbers two

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Abstract

Quasi-symmetric designs with intersection numbers x > 0 and y = x + 2 under the condition λ > 1 are investigated. If D(v, b, r, k, λ; x, y) is a quasi-symmetric design with above conditions then it is shown that either λ = x + 1 or x + 2 or D is a design with the parameters given in the Table 6 or complement of one of these designs.

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References

  1. Bagchi B.: On quasi-symmetric designs. Des. Codes Cryptogr. 2(1), 69–79 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beth T., Jungnickel D., Lenz H.: Design Theory. Cambridge University Press, Cambridge (1986/1999).

  3. Blokhuis A., Calderbank A.R.: Quasi-symmetric designs and the Smith normal form. Des. Codes Cryptogr. 2(2), 189–206 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brouwer A.E., Calderbank A.R.: An Erds-Ko-Rado theorem for regular intersecting families of octads. Graphs Combin. 2(4), 309–316 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  5. Calderbank A.R.: The application of invariant theory to the existence of quasi-symmetric designs. J. Combin. Theory Ser A. 44(1), 94–109 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. Calderbank A.R.: Geometric invariants for quasisymmetric designs. J. Combin. Theory Ser A. 47(1), 101–110 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  7. Calderbank A.R.: Inequalities for quasi-symmetric designs. J. Combin. Theory Ser A. 48(1), 53–64 (1988)

    Article  MathSciNet  Google Scholar 

  8. Calderbank A.R., Frankl P.: Binary codes and quasi-symmetric designs. Discrete Math. 83(2–3), 201–204 (1990)

    Article  MathSciNet  Google Scholar 

  9. Cameron P.J., VanLint J.H.: Designs, Graphs, Codes and their links, London Math., Soc., Students Texts 22. Cambridge University Press, Cambridge (1991).

  10. Goethals J.-M., Seidel J.J.: Strongly regular graphs derived from combinatorial designs. Can. J. Math. 22, 597–614 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  11. Holliday R.L.: Quasi-symmetric block designs with y = λ. Proceedings of the Sixteenth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1985). Congr. Numer. 48, 195–201 (1985).

  12. Maxima: A Computer Algebra System. Version 5.18.1 (2009). http://maxima.sourceforge.net/.

  13. Meyerowitz A.: Quasi-symmetric designs with larger intersection number y. In: Proc. 16th Internat. Conf. Combin., Graph Theory, and Comput. (1986).

  14. Meyerowitz A., Sane S.S., Shrikhande M.S.: New results for quasisymmetric designs—an application of MACSYMA. J. Combin. Theory Ser A. 43(2), 277–290 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  15. Meyerowitz A.: Quasi-symmetric designs with y = λ. J. Combin. Theory Ser A. 59(1), 134–141 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  16. Neumaier A.: Regular sets and quasisymmetric 2-designs. In: Jungnickel D., Vedder K. (eds.) Combinatorial theory (Schloss Rauischholzhausen, 1982), pp. 258–275. Lecture Notes in Math., 969. Springer, Berlin (1982).

  17. Pawale R.M.: Inequalities and bounds for quasi-symmetric 3-designs. J. Combin. Theory Ser A. 60(2), 159–167 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pawale R.M.: Quasi-symmetric designs with fixed difference of block intersection numbers. J. Combin. Des. 15(1), 49–60 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sane S.S., Sane S.S., Sane S.S.: Quasi-symmetric 2,3,4-designs. Combinatorica. 7(3), 291–301 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  20. Tonchev V.D.: Quasi-symmetric 2-(31, 7, 7) designs and a revision of Hamada’s conjecture. J. Combin. Theory Ser A. 42(1), 104–110 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  21. Shrikhande M.S., Sane, S.S.: Quasi-Symmetric Designs. London Math. Society Lecture Notes No. 164. Cambridge University Press, Cambridge (1991).

  22. Witt E.: Über Steinersche Systeme. Abh. Math. Sem. Univ. Hamburg 12, 265–275 (1938)

    Article  Google Scholar 

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Correspondence to Rajendra M. Pawale.

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Communicated by J.D. Key.

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Pawale, R.M. Quasi-symmetric designs with the difference of block intersection numbers two. Des. Codes Cryptogr. 58, 111–121 (2011). https://doi.org/10.1007/s10623-010-9384-x

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  • DOI: https://doi.org/10.1007/s10623-010-9384-x

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