A survey on tight Euclidean t-designs and tight relative t-designs in certain association schemes

  • Eiichi Bannai
  • Etsuko Bannai
  • Yan Zhu


It is known that there is a close analogy between the two relations “Euclidean t-designs vs. spherical t-designs” and “relative t-designs in binary Hamming association schemes vs. combinatorial t-designs.” We first look at this analogy and survey the known results, putting emphasis on the study of tight relative t-designs in certain Q-polynomial association schemes. We then specifically study tight relative 2-designs on two shells in binary Hamming association schemes H(n, 2) and Johnson association schemes J(v, k). The purpose of this paper is to convince the reader that there is a rich theory even for these special cases and that the time is ripe to study tight relative t-designs more systematically for general Q-polynomial association schemes.


Weight Function Orthonormal Basis STEKLOV Institute Association Scheme Cubature Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ei. Bannai, “On tight designs,” Q. J. Math. 28, 433–448 (1977).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ei. Bannai and Et. Bannai, “On Euclidean tight 4-designs,” J. Math. Soc. Japan 58, 775–804 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ei. Bannai and Et. Bannai, “Spherical designs and Euclidean designs,” in Recent Developments in Algebra and Related Areas, Beijing, 2007 (Higher Educ. Press, Beijing, 2009), Adv. Lect. Math. 8, pp. 1–37.Google Scholar
  4. 4.
    Ei. Bannai and Et. Bannai, “A survey on spherical designs and algebraic combinatorics on spheres,” Eur. J. Comb. 30, 1392–1425 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Ei. Bannai and Et. Bannai, “Euclidean designs and coherent configurations,” in Combinatorics and Graphs (Am. Math. Soc., Providence, RI, 2010), Contemp. Math. 531, pp. 59–93.CrossRefGoogle Scholar
  6. 6.
    Ei. Bannai and Et. Bannai, “Remarks on the concepts of t-designs,” J. Appl. Math. Comput. 40 (1–2), 195–207 (2012).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Ei. Bannai and Et. Bannai, “Tight t-designs on two concentric spheres,” Moscow J. Comb. Number Theory 4 (1), 52–77 (2014).zbMATHMathSciNetGoogle Scholar
  8. 8.
    Ei. Bannai, Et. Bannai, and H. Bannai, “On the existence of tight relative 2-designs on binary Hamming association schemes,” Discrete Math. 314, 17–37 (2014).CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ei. Bannai, Et. Bannai, M. Hirao, and M. Sawa, “Cubature formulas in numerical analysis and Euclidean tight designs,” Eur. J. Comb. 31, 423–441 (2010).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Ei. Bannai, Et. Bannai, S. Suda, and H. Tanaka, “On relative t-designs in polynomial association schemes,” arXiv: 1303.7163 [math.CO].Google Scholar
  11. 11.
    Ei. Bannai and R. M. Damerell, “Tight spherical designs. I,” J. Math. Soc. Japan 31, 199–207 (1979).CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ei. Bannai and R. M. Damerell, “Tight spherical designs. II,” J. London Math. Soc., Ser. 2, 21, 13–30 (1980).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Ei. Bannai and T. Ito, Algebraic Combinatorics. I: Association Schemes (Benjamin/Cummings, Menlo Park, CA, 1984).Google Scholar
  14. 14.
    Ei. Bannai, A. Munemasa, and B. Venkov, “The nonexistence of certain tight spherical designs,” Algebra Anal. 16 (4), 1–23 (2004) [St. Petersburg Math. J. 16, 609–625 (2005)].MathSciNetGoogle Scholar
  15. 15.
    Ei. Bannai and N. J. A. Sloane, “Uniqueness of certain spherical codes,” Can. J. Math. 33, 437–449 (1981).CrossRefMathSciNetGoogle Scholar
  16. 16.
    Et. Bannai, “On antipodal Euclidean tight (2e + 1)-designs,” J. Algebr. Comb. 24, 391–414 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Et. Bannai, “New examples of Euclidean tight 4-designs,” Eur. J. Comb. 30, 655–667 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    T. Beth, D. Jungnickel, and H. Lenz, Design Theory (Bibliogr. Inst., Mannheim, 1985).zbMATHGoogle Scholar
  19. 19.
    A. Bremner, “A Diophantine equation arising from tight 4-designs,” Osaka J. Math. 16, 353–356 (1979).zbMATHMathSciNetGoogle Scholar
  20. 20.
    A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer, Berlin, 1989).CrossRefzbMATHGoogle Scholar
  21. 21.
    R. Calderbank and W. M. Kantor, “The geometry of two-weight codes,” Bull. London Math. Soc. 18, 97–122 (1986).CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and Their Links (Cambridge Univ. Press, Cambridge, 1991), London Math. Soc. Stud. Texts 22.CrossRefzbMATHGoogle Scholar
  23. 23.
    P. J. Cameron and J. J. Seidel, “Quadratic forms over GF(2),” Indag. Math. 35, 1–8 (1973).CrossRefMathSciNetGoogle Scholar
  24. 24.
    P. Delsarte, “An algebraic approach to the association schemes of the coding theory,” Thesis (Univ. Cathol. Louvain, Louvain-la-Neuve, 1973); An Algebraic Approach to the Association Schemes of the Coding Theory (Historical Jrl., Ann Arbor, MI, 1973), Philips Res. Rep., Suppl. 10.zbMATHGoogle Scholar
  25. 25.
    P. Delsarte, “Association schemes and t-designs in regular semilattices,” J. Comb. Theory A 20, 230–243 (1976).CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    P. Delsarte, “Pairs of vectors in the space of an association scheme,” Philips Res. Rep. 32, 373–411 (1977).MathSciNetGoogle Scholar
  27. 27.
    P. Delsarte, J. M. Goethals, and J. J. Seidel, “Bounds for systems of lines, and Jacobi polynomials,” Philips Res. Rep. 30, 91–105 (1975).zbMATHGoogle Scholar
  28. 28.
    P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs,” Geom. Dedicata 6, 363–388 (1977).CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    P. Delsarte and J. J. Seidel, “Fisher type inequalities for Euclidean t-designs,” Linear Algebra Appl. 114–115, 213–230 (1989).CrossRefMathSciNetGoogle Scholar
  30. 30.
    P. Dukes and J. Short-Gershman, “Nonexistence results for tight block designs,” J. Algebr. Comb. 38, 103–119 (2013).CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    H. Enomoto, N. Ito, and R. Noda, “Tight 4-designs,” Osaka J. Math. 16, 39–43 (1979).zbMATHMathSciNetGoogle Scholar
  32. 32.
    S. G. Hoggar, “t-Designs in projective spaces,” Eur. J. Comb. 3, 233–254 (1982).CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    D. R. Hughes, “On t-designs and groups,” Am. J. Math. 87, 761–778 (1965).CrossRefzbMATHGoogle Scholar
  34. 34.
    P. Keevash, “The existence of designs,” arXiv: 1401.3665 [math.CO].Google Scholar
  35. 35.
    G. Kuperberg, “Special moments,” Adv. Appl. Math. 34, 853–870 (2005).CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    G. Kuperberg, S. Lovett, and R. Peled, “Probabilistic existence of regular combinatorial structures,” arXiv: 1302.4295 [math.CO].Google Scholar
  37. 37.
    V. I. Levenshtein, “Designs as maximum codes in polynomial metric spaces,” Acta Appl. Math. 29, 1–82 (1992).CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Z. Li, Ei. Bannai, and Et. Bannai, “Tight relative 2- and 4-designs on binary Hamming association schemes,” Graphs Comb. 30, 203–227 (2014).CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    W. J. Martin, “Mixed block designs,” J. Comb. Des. 6, 151–163 (1998).CrossRefzbMATHGoogle Scholar
  40. 40.
    W. J. Martin, “Designs in product association schemes,” Des. Codes Cryptogr. 16, 271–289 (1999).CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    W. J. Martin, “Symmetric designs, sets with two intersection numbers, and Krein parameters of incidence graphs,” J. Comb. Math. Comb. Comput. 38, 185–196 (2001).zbMATHGoogle Scholar
  42. 42.
    R. L. McFarland, “A family of difference sets in non-cyclic groups,” J. Comb. Theory A 15, 1–10 (1973).CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    G. Nebe and B. Venkov, “On tight spherical designs,” Algebra Anal. 24 (3), 163–171 (2012) [St. Petersburg Math. J. 24, 485–491 (2013)].MathSciNetGoogle Scholar
  44. 44.
    A. Neumaier, “Combinatorial configurations in terms of distances,” Memorandum 81-09 (Dept. Math., Eindhoven Univ. Technol., Eindhoven, 1981).Google Scholar
  45. 45.
    A. Neumaier and J. J. Seidel, “Discrete measures for spherical designs, eutactic stars and lattices,” Indag. Math. 50, 321–334 (1988).CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    C. Peterson, “On tight 6-designs,” Osaka J. Math. 14, 417–435 (1977).zbMATHMathSciNetGoogle Scholar
  47. 47.
    A. Ya. Petrenyuk, “Fisher’s inequality for tactical configurations,” Mat. Zametki 4 (4), 417–424 (1968) [Math. Notes 4, 742–746 (1968)].MathSciNetGoogle Scholar
  48. 48.
    D. K. Ray-Chaudhuri and R. M. Wilson, “On t-designs,” Osaka J. Math. 12, 737–744 (1975).zbMATHMathSciNetGoogle Scholar
  49. 49.
    P. D. Seymour and T. Zaslavsky, “Averaging sets: A generalization of mean values and spherical designs,” Adv. Math. 52, 213–240 (1984).CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    L. Teirlinck, “Non-trivial t-designs without repeated blocks exist for all t,” Discrete Math. 65, 301–311 (1987).CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    W. D. Wallis, “Construction of strongly regular graphs using affine designs,” Bull. Aust. Math. Soc. 4, 41–49 (1971); 5, 431 (1971).CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    D. R. Woodall, “Square λ-linked designs,” Proc. London Math. Soc., Ser. 3, 20, 669–687 (1970).CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    Z. Xiang, “A Fisher type inequality for weighted regular t-wise balanced designs,” J. Comb. Theory A 119, 1523–1527 (2012).CrossRefzbMATHGoogle Scholar
  54. 54.
    Y. Zhu, E. Bannai, and E. Bannai, “Tight relative 2-designs on two shells in Johnson association schemes,” submitted to Discrete Math.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina
  2. 2.FukuokaJapan

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