Advertisement

Journal of Algebraic Combinatorics

, Volume 47, Issue 2, pp 301–318 | Cite as

Nonexistence of nontrivial tight 8-designs

  • Ziqing Xiang
Article
  • 165 Downloads

Abstract

Tight t-designs are t-designs whose sizes achieve the Fisher type lower bound. We give a new necessary condition for the existence of nontrivial tight designs and then use it to show that there do not exist nontrivial tight 8-designs.

Keywords

Tight design Intersection number Product of consecutive integers 

Notes

Acknowledgements

It is my great pleasure to thank Eiichi Bannai for introducing me to the subject and suggesting this problem to me. I also thank Etsuko Bannai, Dino Lorenzini, Yaokun Wu and Jiacheng Xia for useful comments and discussions. I gratefully acknowledge financial support from the Office of the Vice President for Research at the University of Georgia, and from the Research and Training Group in Algebraic Geometry, Algebra and Number Theory.

References

  1. 1.
    Bannai, E.: On tight designs. Q. J. Math. 28(4), 433–448 (1977). doi: 10.1093/qmath/28.4.433 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bannai, E., and Ito, T.: Unpublished (1977)Google Scholar
  3. 3.
    Bremner, A.: A diophantine equation arising from tight \(4\)-designs. Osaka J. Math. 16(2), 353–356 (1979). http://hdl.handle.net/11094/8744
  4. 4.
    Delsarte, P.: An algebraic approach to the association schemes of coding theory. Philips Res. Repts. Suppl. 10 (1973)Google Scholar
  5. 5.
    Dukes, P., Short-Gershman, J.: Nonexistence results for tight block designs. J. Algebr. Comb. 38(1), 103–119 (2013). doi: 10.1007/s10801-012-0395-8 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Edgar, T., Spivey, M.Z.: Multiplicative functions generalized binomial coefficients, and generalized catalan numbers. J. Integer Seq. 19(2), 737–744 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hall Jr., M.: Combinatorial Theory, 2nd edn. Wiley, New York (1998)zbMATHGoogle Scholar
  8. 8.
    Hilliker, D.L., Straus, E.G.: Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge’s theorem. Trans. Am. Math. Soc. 280(2), 637–657 (1983). doi: 10.1090/S0002-9947-1983-0716842-3 MathSciNetzbMATHGoogle Scholar
  9. 9.
    Hindry, M., and Silverman, J.H.: Diophantine Geometry: An Introduction (2000). http://www.springer.com/us/book/9780387989754
  10. 10.
    Lorenzini, D., and Xiang, Z.: Integral points on variable separated curves (2015)Google Scholar
  11. 11.
    Peterson, C.: On tight \(6\)-designs. Osaka J. Math. 14(2), 417–435 (1977). http://hdl.handle.net/11094/3990
  12. 12.
    Ray-Chaudhuri, D.K., and Wilson, R.M.: On \(t\)-designs. Osaka J. Math. 12(3), 737–744 (1975). http://hdl.handle.net/11094/7296
  13. 13.
    Runge, C.: Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen. Journal für die reine und angewandte Mathematik, 100, 425–435 (1887). http://eudml.org/doc/148675
  14. 14.
    Stroeker, R.J.: On the diophantine equation \((2y^2-3)^2 = x^2(3x^2-2)\) in connection with the existence of non-trivial tight \(4\)-designs. Indag. Math. 84(3), 353–358 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Walsh, P.G.: A quantitative version of Runge’s theorem on Diophantine equations. Acta Arith. 62(2), 157–172 (1992). http://eudml.org/doc/206487
  16. 16.
    Zannier, U.: Lecture Notes on Diophantine Analysis (2015)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA

Personalised recommendations