Abstract
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.
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References
Ei. Bannai, “On tight designs,” Q. J. Math. 28, 433–448 (1977).
Ei. Bannai and Et. Bannai, “On Euclidean tight 4-designs,” J. Math. Soc. Japan 58, 775–804 (2006).
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.
Ei. Bannai and Et. Bannai, “A survey on spherical designs and algebraic combinatorics on spheres,” Eur. J. Comb. 30, 1392–1425 (2009).
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.
Ei. Bannai and Et. Bannai, “Remarks on the concepts of t-designs,” J. Appl. Math. Comput. 40 (1–2), 195–207 (2012).
Ei. Bannai and Et. Bannai, “Tight t-designs on two concentric spheres,” Moscow J. Comb. Number Theory 4 (1), 52–77 (2014).
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).
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).
Ei. Bannai, Et. Bannai, S. Suda, and H. Tanaka, “On relative t-designs in polynomial association schemes,” arXiv: 1303.7163 [math.CO].
Ei. Bannai and R. M. Damerell, “Tight spherical designs. I,” J. Math. Soc. Japan 31, 199–207 (1979).
Ei. Bannai and R. M. Damerell, “Tight spherical designs. II,” J. London Math. Soc., Ser. 2, 21, 13–30 (1980).
Ei. Bannai and T. Ito, Algebraic Combinatorics. I: Association Schemes (Benjamin/Cummings, Menlo Park, CA, 1984).
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)].
Ei. Bannai and N. J. A. Sloane, “Uniqueness of certain spherical codes,” Can. J. Math. 33, 437–449 (1981).
Et. Bannai, “On antipodal Euclidean tight (2e + 1)-designs,” J. Algebr. Comb. 24, 391–414 (2006).
Et. Bannai, “New examples of Euclidean tight 4-designs,” Eur. J. Comb. 30, 655–667 (2009).
T. Beth, D. Jungnickel, and H. Lenz, Design Theory (Bibliogr. Inst., Mannheim, 1985).
A. Bremner, “A Diophantine equation arising from tight 4-designs,” Osaka J. Math. 16, 353–356 (1979).
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer, Berlin, 1989).
R. Calderbank and W. M. Kantor, “The geometry of two-weight codes,” Bull. London Math. Soc. 18, 97–122 (1986).
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.
P. J. Cameron and J. J. Seidel, “Quadratic forms over GF(2),” Indag. Math. 35, 1–8 (1973).
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.
P. Delsarte, “Association schemes and t-designs in regular semilattices,” J. Comb. Theory A 20, 230–243 (1976).
P. Delsarte, “Pairs of vectors in the space of an association scheme,” Philips Res. Rep. 32, 373–411 (1977).
P. Delsarte, J. M. Goethals, and J. J. Seidel, “Bounds for systems of lines, and Jacobi polynomials,” Philips Res. Rep. 30, 91–105 (1975).
P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs,” Geom. Dedicata 6, 363–388 (1977).
P. Delsarte and J. J. Seidel, “Fisher type inequalities for Euclidean t-designs,” Linear Algebra Appl. 114–115, 213–230 (1989).
P. Dukes and J. Short-Gershman, “Nonexistence results for tight block designs,” J. Algebr. Comb. 38, 103–119 (2013).
H. Enomoto, N. Ito, and R. Noda, “Tight 4-designs,” Osaka J. Math. 16, 39–43 (1979).
S. G. Hoggar, “t-Designs in projective spaces,” Eur. J. Comb. 3, 233–254 (1982).
D. R. Hughes, “On t-designs and groups,” Am. J. Math. 87, 761–778 (1965).
P. Keevash, “The existence of designs,” arXiv: 1401.3665 [math.CO].
G. Kuperberg, “Special moments,” Adv. Appl. Math. 34, 853–870 (2005).
G. Kuperberg, S. Lovett, and R. Peled, “Probabilistic existence of regular combinatorial structures,” arXiv: 1302.4295 [math.CO].
V. I. Levenshtein, “Designs as maximum codes in polynomial metric spaces,” Acta Appl. Math. 29, 1–82 (1992).
Z. Li, Ei. Bannai, and Et. Bannai, “Tight relative 2- and 4-designs on binary Hamming association schemes,” Graphs Comb. 30, 203–227 (2014).
W. J. Martin, “Mixed block designs,” J. Comb. Des. 6, 151–163 (1998).
W. J. Martin, “Designs in product association schemes,” Des. Codes Cryptogr. 16, 271–289 (1999).
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).
R. L. McFarland, “A family of difference sets in non-cyclic groups,” J. Comb. Theory A 15, 1–10 (1973).
G. Nebe and B. Venkov, “On tight spherical designs,” Algebra Anal. 24 (3), 163–171 (2012) [St. Petersburg Math. J. 24, 485–491 (2013)].
A. Neumaier, “Combinatorial configurations in terms of distances,” Memorandum 81-09 (Dept. Math., Eindhoven Univ. Technol., Eindhoven, 1981).
A. Neumaier and J. J. Seidel, “Discrete measures for spherical designs, eutactic stars and lattices,” Indag. Math. 50, 321–334 (1988).
C. Peterson, “On tight 6-designs,” Osaka J. Math. 14, 417–435 (1977).
A. Ya. Petrenyuk, “Fisher’s inequality for tactical configurations,” Mat. Zametki 4 (4), 417–424 (1968) [Math. Notes 4, 742–746 (1968)].
D. K. Ray-Chaudhuri and R. M. Wilson, “On t-designs,” Osaka J. Math. 12, 737–744 (1975).
P. D. Seymour and T. Zaslavsky, “Averaging sets: A generalization of mean values and spherical designs,” Adv. Math. 52, 213–240 (1984).
L. Teirlinck, “Non-trivial t-designs without repeated blocks exist for all t,” Discrete Math. 65, 301–311 (1987).
W. D. Wallis, “Construction of strongly regular graphs using affine designs,” Bull. Aust. Math. Soc. 4, 41–49 (1971); 5, 431 (1971).
D. R. Woodall, “Square λ-linked designs,” Proc. London Math. Soc., Ser. 3, 20, 669–687 (1970).
Z. Xiang, “A Fisher type inequality for weighted regular t-wise balanced designs,” J. Comb. Theory A 119, 1523–1527 (2012).
Y. Zhu, E. Bannai, and E. Bannai, “Tight relative 2-designs on two shells in Johnson association schemes,” submitted to Discrete Math.
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Bannai, E., Bannai, E. & Zhu, Y. A survey on tight Euclidean t-designs and tight relative t-designs in certain association schemes. Proc. Steklov Inst. Math. 288, 189–202 (2015). https://doi.org/10.1134/S0081543815010149
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DOI: https://doi.org/10.1134/S0081543815010149