Nonexistence of nontrivial tight 8-designs



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.


Tight design Intersection number Product of consecutive integers 



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.


  1. 1.
    Bannai, E.: On tight designs. Q. J. Math. 28(4), 433–448 (1977). doi: 10.1093/qmath/28.4.433 MathSciNetCrossRefMATHGoogle 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    Bannai, E., and Ito, T.: Unpublished (1977)Google Scholar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).
  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 MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  7. 7.
    Hall Jr., M.: Combinatorial Theory, 2nd edn. Wiley, New York (1998)MATHGoogle 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 MathSciNetMATHGoogle Scholar
  9. 9.
    Hindry, M., and Silverman, J.H.: Diophantine Geometry: An Introduction (2000).
  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).
  12. 12.
    Ray-Chaudhuri, D.K., and Wilson, R.M.: On \(t\)-designs. Osaka J. Math. 12(3), 737–744 (1975).
  13. 13.
    Runge, C.: Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen. Journal für die reine und angewandte Mathematik, 100, 425–435 (1887).
  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)CrossRefMATHGoogle Scholar
  15. 15.
    Walsh, P.G.: A quantitative version of Runge’s theorem on Diophantine equations. Acta Arith. 62(2), 157–172 (1992).
  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