A survey on spherical designs and algebraic combinatorics on spheres

European Journal of Combinatorics

Volumn 30, Issue 6, 2009, Pages 1392-1425

  Bannai, Eiichi ^a   Bannai, Etsuko ^b  

^a KYUSHU UNIVERSITY   (Japan)

^b Misakigaoka 2 8 21   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 68649091969     PISSN: 01956698     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ejc.2008.11.007     Document Type: Article

References (185)

1. Abdukhalikov, Bannai E., and Suda. Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets. J. Combin. Theory (A) 116 (2009) 434-448
2. Arias de Reyna J. A generalized mean-value theorem. Mh. Math. 106 (1988) 95-97
3. Assmus Jr. E.F., and Mattson Jr. H.F. New 5-designs. J. Combin. Theory 6 (1969) 122-151
4. Bachoc C. On harmonic weight enumerators of binary codes. Designs and Codes-A Memorial Tribute to Ed Assmus. Des. Codes Cryptogr. 18 1-3 (1999) 11-28
5. Bachoc C. Linear programming bounds for codes in Grassmannian spaces. IEEE Trans. Inform. Theory 52 5 (2006) 2111-2125
6. Bachoc C. Designs, groups and lattices. J. Theor. Nombres Bordeaux 17 1 (2005) 25-44
7. Bachoc C., Bannai E., and Coulangeon R. Codes and designs in Grassmannian spaces. Discrete Math. 277 1-3 (2004) 15-28
8. Bachoc C., Coulangeon R., and Nebe G. Designs in Grassmannian spaces and lattices. J. Algebraic Combin. 16 1 (2002) 5-19
9. Bachoc C., and Vallentin F. New upper bounds for kissing numbers from semidefinite programming. J. Amer. Math. Soc. 21 3 (2008) 909-924
10. Bachoc C., and Vallentin F. Optimality and uniqueness of the (4, 10, 1/6) spherical code. J. Combin. Theory Ser. A 116 (2009) 195-204
11. C. Bachoc, F. Vallentin, Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps, J. European Combin., in press (doi:10.1016/j.ejc.2008.07.017)
12. Bachoc C., and Venkov B. Modular forms, lattices and spherical designs. Reseaux euclidiens, designs spheriques et formes modulaires. Monogr. Enseign. Math. vol. 37 (2001), Enseignement Math., Geneva 87-111
13. Bajnok B. Constructions of spherical 3-designs. Graphs Combin. 14 (1998) 97-107
14. Bajnok B. Chebyshev-type quadrature formulas on the sphere. Proceedings of the Twenty-second Southeastern Conference on Combinatorics, Graph Theory, and Computing. Congr. Numer. vol. 85 (1991) 214-218
15. Bajnok B. Construction of designs on the 2-sphere. European J. Combin. 12 5 (1991) 377-382
16. Bajnok B. Construction of spherical 4- and 5-designs. Graphs Combin. 7 3 (1991) 219-233
17. Bajnok B. Construction of spherical t-designs. Geom. Dedicata 43 2 (1992) 167-179
18. Bajnok B. Orbits of the hyperoctahedral group as Euclidean designs. J. Algebraic Combin. 25 (2007) 357-471
19. Bajnok B. On Euclidean designs. Adv. Geom. 6 (2006) 423-438
20. B. Ballinger, G. Blekherman, H. Cohn, N. Giansiracusa, E. Kelly, A. Schuermann, Experimental study of energy-minimizing point configurations on spheres, Exp. Math. (in press)
21. S. Bang, J.H. Koolen, V. Moulton, (in preparation)
22. Bannai E. On some spherical t-designs. J. Combin. Theory Ser. A 26 2 (1979) 157-161
23. Bannai E. Spherical t-designs which are orbits of finite groups. J. Math. Soc. Japan 36 (1984) 341-354
24. Bannai E. Spherical t-designs and group representations. Contemp. Math. 34 (1984) 85-107
25. Bannai E. Rigid spherical t-designs and a theorem of Y. Hong. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 3 (1987) 485-489
26. Bannai E. On antipodal Euclidean tight (2 e + 1)-designs. J. Algebraic Combin. 24 (2006) 391-414
27. E. Bannai, New examples of Euclidean tight 4-designs, European J. Combin., in press (doi:10.1016/j.ejc.2008.07.012)
28. Bannai E. On extremal finite set in the sphere and other metric spaces. Algebraic, Extremal and Metric Combinatorics. Lond. Math. Soc. Lect. Note Ser. vol. 131 (1988) 13-38
29. Bannai E., and Bannai E. Algebraic Combinatorics on Spheres (1999), Springer, Tokyo (in Japanese)
30. E. Bannai, E. Bannai, On antipodal spherical t-designs of degree s with t ≥ 2 s - 3, J. Combin. Inform. Syst. Sci., (in press) Honoring the 75-th birthday of Prof. D. K. Ray-Chaudhuri
31. Bannai E., and Bannai E. Spherical designs and Euclidean designs. Recent Developments in Algebra and Related Areas. ALM vol. 8 (2008), Higher Education Press and International Press, Beijing, Boston 1-37
32. 1 = 3. Contrib. Discrete Math. 1 1 (2006) 68-79
33. Bannai E., and Bannai E. A note on the spherical embeddings of strongly regular graphs. European J. Combin. 26 (2005) 1177-1179
34. Bannai E., and Bannai E. On Euclidean tight 4-designs. J. Math. Soc. Japan 58 (2006) 775-804
35. E. Bannai, E. Bannai, Euclidean designs and coherent configurations (in preparation)
36. Bannai E., and Bannai E. Tight Gaussian 4-designs. J. Algebraic Combin. 22 1 (2005) 39-63
37. Bannai E., Bannai E., and Bannai H. Uniqueness of certain association schemes. European J. Combin. 29 6 (2008) 1379-1395
38. E. Bannai, E. Bannai, M. Hirao, M. Sawa, Cubature formulas in numerical analysis and Euclidean tight designs, (in press) in a special issue in honor of Michel Deza (European. J. Combin.)
39. Bannai E., Bannai E., and Stanton D. An upper bound for the cardinality of an s-distance subset in real Euclidean space. II. Combinatorica 3 2 (1983) 147-152
40. Bannai E., Bannai E., and Suprijanto D. On the strong non-rigidity of certain tight Euclidean designs. European J. Combin. 28 (2007) 1662-1680
41. Bannai E., Blokhuis A., Delsarte Ph., and Seidel J.J. An addition formula for hyperbolic space. J. Combin. Theory Ser. A 36 3 (1984) 332-341
42. Bannai E., and Damerell R.M. Tight spherical designs. I. J. Math. Soc. Japan 31 1 (1979) 199-207
43. Bannai E., and Damerell R.M. Tight spherical designs. II. J. London Math. Soc. (2) 21 1 (1980) 13-30
44. Bannai E., and Hoggar S.G. On tight t-designs in compact symmetric spaces of rank one. Proc. Japan Acad. Ser. A Math. Sci. 61 3 (1985) 78-82
45. Bannai E., and Hoggar S.G. Tight t-designs and squarefree integers. European J. Combin. 10 2 (1989) 113-135
46. Bannai E., and Ito T. Algebraic Combinatorics I: Association Schemes (1984), Benjamin/Cummings, Menlo Park, CA
47. Bannai E., Koike M., Shinohara M., and Tagami M. Spherical designs attached to extremal lattices and the modulo p property of Fourier coefficients of extremal modular forms. Mosc. Math. J. 6 2 (2006) 225-264
48. Bannai E., Munemasa A., and Venkov B. The nonexistence of certain tight spherical designs. Algebra Anal. 16 (2004) 1-23
49. Bannai E., and Sloane N.J.A. Uniqueness of certain spherical codes. Canad. J. Math. 33 2 (1981) 437-449
50. Barg A., and Musin O.R. Codes in spherical caps. Adv. Math. Commun. 1 1 (2007) 131-149
51. Barg A., and Nogin D. A bound on Grassmannian codes. J. Combin. Theory Ser. A 113 8 (2006) 1629-1635
52. Bernstein S.N. On quadrature formulas with positive coefficients. Izv. Akad. Nauk. SSSR, Ser. Mat. 1 4 (1937) 479-503 (in Russian). (Reprinted in collected works, Vol 2, Izdat. Akad. Nauk SSSR, Moskow, (1954) 205-227.) See also the announcements in C. R. Acad. Sci. Paris 204 (1937) 1294-1296; 1526-1529
53. Bernstein S.N. Sur les formulas de quadrature de cotes et Tchebycheff. C. R. Acad. Sci. URSS (Dokl. Akad. Nauk, SSSR), N. S. 14 (1937) 323-327
54. A. Blokhuis, Few-distance sets, CWI Tract, 7. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1984, iv+70 pp.
55. p, q. Symposia Mathematica, Vol. XXVIII. Sympos. Math. vol. XXVIII (1986), Academic Press, London 145-158
56. Bourbaki N. Groups et Algebre de Lie (1968), Hermann, Paris (Chapter 4-6)
57. Boyvalenkov P. Extremal polynomials for obtaining bounds for spherical codes and designs. Discrete Comput. Geom. 14 (1995) 167-188
58. Boyvalenkov P., Boumova S., and Danev D. Necessary conditions for existence of some designs in polynomial metric spaces. European J. Combin. 20 3 (1999) 213-225
59. Boyvalenkov P., and Danev D. Uniqueness of the 120-point spherical 11-design in four dimensions. Arch. Math. (Basel) 77 (2001) 360-368
60. Boyvalenkov P., Danev D., and Kazakov P. Indices of spherical codes. DIMACS Ser. Discrete Math. Comput. Sci. 58 (2001) 47-57
61. Boyvalenkov P., Danev D., and Landgev I. On maximal spherical codes. II. J. Combin. Des. 7 5 (1999) 316-326
62. Boyvalenkov P., Danev D., and Nikova S. Nonexistence of certain spherical designs of odd strengths and cardinalities. Discrete Comput. Geom. 21 1 (1999) 143-156
63. Boyvalenkov P., and Nikova S. On lower bounds on the size of designs in compact symmetric spaces of rank 1. Arch. Math. (Basel) 68 1 (1997) 81-88
64. Brouwer A.E., Cohen A.M., and Neumaier A. Distance-regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics and Related Areas (3)) vol. 18 (1989), Springer-Verlag, Berlin xviii+495 pp.
65. 4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. Proc. London Math. Soc. (3) 75 2 (1997) 436-480
66. Calderbank A.R., Hardin R.H., Rains E.M., Shor P.W., and Sloane N.J.A. A group-theoretic framework for the construction of packings in Grassmannian spaces. J. Algebraic Combin. 9 2 (1999) 129-140
67. Calderbank A.R., Rains E.M., Shor P.W., and Sloane N.J.A. Quantum error correction and orthogonal geometry. Phys. Rev. Lett. 78 3 (1997) 405-408
68. Calderbank A.R., Rains E.M., Shor P.W., and Sloane N.J.A. Quantum error correction via codes over GF (4). IEEE Trans. Inform. Theory 44 4 (1998) 1369-1387
69. Cameron P.J., Goethals J.-M., and Seidel J.J. The Krein condition, spherical designs, Norton algebras and permutation groups. Nederl. Akad. Wetensch. Indag. Math. 40 2 (1978) 196-206
70. Cameron P.J., and Seidel J.J. Quadratic forms over G F (2). Nederl. Akad. Wetensch. Proc. Ser. A 76 (1973) 1-8 Indag. Math. 35
71. Cameron P.J., and van Lint J.H. Designs, Graphs, Codes and their Links. London Math. Soc. Student Texts vol. 22 (1991), Cambridge Univ. Press
72. Chen X., and Womersley R.S. Existence of solutions to systems of underdetermined equations and spherical designs. SIAM J. Numer. Anal. 44 6 (2006) 2326-2341
73. 4 root system is not universally optimal. Exp. Math. 16 3 (2007) 313-320
74. Cohn H., and Elkies N. New upper bounds on sphere packings. I. Ann. of Math. (2) 157 2 (2003) 689-714
75. H. Cohn, A. Kumar, Optimality and uniqueness of the Leech lattice among lattices, Ann. of Math. (in press)
76. Cohn H., and Kumar A. The densest lattice in twenty-four dimensions. Electron. Res. Announc. Amer. Math. Soc. 10 (2004) 58-67
77. Cohn H., and Kumar A. Universally optimal distribution of points on spheres. J. Amer. Math. Soc. 20 (2007) 99-148
78. Cohn H., and Kumar A. Uniqueness of the (22, 891, 1/4) spherical code. New York J. Math. 13 (2007) 147-157
79. Conway J.H., Hardin R.H., and Sloane N.J.A. Packing lines, planes, etc.: Packings in Grassmannian spaces. Exp. Math. 5 (1996) 139-159
80. Conway J.H., and Sloane N.J.A. Sphere Packings, Lattices and Groups (With additional contributions by E. Bannai, R.E. Borcherds, J. Leech, S.P. Norton, A.M. Odlyzko, R.A. Parker, L. Queen and B.B. Venkov). third edition. Grundlehren der Mathematischen Wissenschaften vol. 290 (1999), Springer-Verlag, New York lxxiv+703 pp.
81. Coulangeon R. Spherical designs and zeta functions of lattices. Int. Math. Res. Not. (2006) Art. ID 49620, 16 pp.
82. 7. Des. Codes Cryptogr. 34 2-3 (2005) 333-337
83. de Caen D., and van Dam E.R. Association Schemes Related to Kasami Codes and Kerdock Sets. Des. Codes Crypt. 18 1-3 (1999) 89-102
84. de la Harpe P., and Pache C. Spherical designs and finite group representations (some results of E. Bannai). European J. of Combin. 25 (2004) 213-227
85. de la Harpe P., and Pache C. Cubature formulas, geometrical designs, reproducing kernels, and Markov operators. Infinite Groups: Geometric, Combinatorial and Dynamical Aspects. Progr. Math. vol. 248 (2005), Birkhauser, Basel 219-267
86. de la Harpe P., Pache C., and Venkov B. Construction of spherical cubature formulas using lattices. Algebra Anal. 18 1 (2006) 162-186 Translation in St. Petersburg Math. J. 18 (1) (2007) 119-139
87. Delsarte P. An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10 (1973)
88. Delsarte P., Goethals J.M., and Seidel J.J. Spherical codes and designs. Geom. Dedicata 6 (1977) 363-388
89. Delsarte P., Goethals J.M., and Seidel J.J. Bounds for systems of lines and Jacobi polynomials. Philip. Res. Rep. 30 (1975) Boukamo Volume 91*-105*
90. Delsarte P., and Levenshtein V.I. Association schemes and coding theory. Information Theory: 1948-1998. IEEE Trans. Inform. Theory 44 6 (1998) 2477-2504
91. Delsarte P., and Seidel J.J. Fisher type inequalities for Euclidean t-designs. Linear Algebra Appl. 114-115 (1989) 213-230
92. Deza M., and Frankl P. Bounds on the maximum number of vectors with given scalar products. Proc. Amer. Math. Soc. 95 2 (1985) 323-329
93. Dorofeev A.YA., Kazarin L.S., Sidelnikov V.M., and Tuzhilin M.E. Matrix groups related to the quaternion group and spherical orbit codes. Des. Codes Cryptogr. 37 (2005) 391-404
94. Ericson T., and Zinoviev V. Codes on Euclidean spheres. North-Holland Mathematical Library vol. 63 (2001), North-Holland Publishing Co., Amsterdam xiv+549 pp.
95. Gijswijt D., Schrijver A., and Tanaka H. New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming. J. Combin. Theory Ser. A 113 (2006) 1719-1731
96. Godsil C.D. Algebraic Combinatorics. Chapman and Hall Mathematics Series (1993), Chapman and Hall, New York xvi+362 pp.
97. J.-M. Goethals, J.J. Seidel, Spherical designs, in: Proc. Symposium in Pure Math., vol. 34, 1979, pp. 255-272
98. Goethals J.-M., and Seidel J.J. Cubature formulae, polytopes and spherical designs. In: Davis C., Grünbaum B., and Sherk F.A. (Eds). The Geometric Vein: The Coxter Festschrift (1981), Springer-Verlag 203-218
99. Griess Jr. R.L. Few-cosine spherical codes and Barnes-Wall lattices. J. Combin. Theory (A) 115 (2008) 1211-1234
100. Hales T.C. A proof of the Kepler conjecture. Ann. of Math. (2) 162 3 (2005) 1065-1185
101. Hardin R.H., and Sloane N.J.A. New spherical 4-designs. Discrete Math. 106-107 (1992) 255-264
102. Hardin R.H., and Sloane N.J.A. McLaren's improved snub cube and other new spherical designs in three dimensions. Discrete Comput. Geom. 15 4 (1996) 429-441
103. Helgason S. Differential Geometry, Lie Groups, and Symmetric Spaces (1978), Academic Press, New York, London
104. Hoggar S.G. t-designs in projective spaces. European J. Combin. 3 3 (1982) 233-254
105. d - 1. European J. Combin. 5 1 (1984) 29-36
106. Hoggar S.G. t-designs in Delsarte spaces. Coding Theory and Design Theory, Part II. IMA Vol. Math. Appl. vol. 21 (1990), Springer, New York 144-165
107. 2. European J. Combin. 3 3 (1982) 255-258
108. Janusz G.T. Overlap and covering polynomials with applications to designs and self-dual codes. SIAM J. Discrete Math. 13 (2000) 154-178
109. Kabatiansky, and Lebenshtein. Bounds for packings on sphere and in space. Probl. Inform. Transm. 14 (1978) 1-17
110. Koch H.V. Unimoduler lattices and self-dual codes. Proceedings of the International Congress of Mathematicians, vol. 1, 2 (1987), Amer. Math. Soc., Providence, RI 457-465
111. Koch H.V. On self-dual doubly-even extremal codes. Discrete Math. 83 (1990) 291-300
112. Korevaar J. Chebychev-type quadratures: Use of complex analysis and potential theory (notes by A.B.J. Kuijlaars). In: Gauthier P.M., and Sabidussi G. (Eds). Complex Potential Theory (1994), Kluwer 325-364
113. Korevaar J., and Meyers J.L.H. Spherical faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere. J. Integral Transforms Special Funct. 1 (1993) 105-117
114. Kuijlaars A.B.J. The minimal number of nodes in Chebyshev type quadrature formulas. Indag. Math. 4 (1993) 339-362
115. Kuperberg G. Special moments. Adv. Appl. Math. 34 4 (2005) 853-870
116. Kuperberg G. Numerical cubature using error-correcting codes. SIAM J. Numer. Anal. 44 3 (2006) 897-907
117. Kuperberg G. Numerical cubature from Archimedes' hat-box theorem. SIAM J. Numer. Anal. 44 3 (2006) 908-935
118. Larman D.G., Rogers C.A., and Seidel J.J. On two-distance sets in Euclidean space. Bull. London Math. Soc. 9 (1977) 261-267
119. Leech J. Equiliblium of sets of particles on a sphere. Math. Gaz. 41 (1957) 81-90
120. Lehmer D.H. The vanishing of Ramanujans function τ (n). Duke Math. J. 14 (1947) 429-433
121. Lempken W., Schroder B., and Tiep P.H. Symmetric squares, spherical designs, and lattice minima. With an appendix by Christine Bachoc and Tiep. J. Algebra 240 1 (2001) 185-208
122. Levenshtein V.I. On bounds for packing in n-dimensional Euclidean space. Soviet Math. Dokl. 20 (1979) 417-421
123. Levenshtein V.I. Universal bounds for codes and designs. Handbook of Coding Theory, vol. I, II (1998), North-Holland, Amsterdam 499-648
124. Levenshtein V.I. Designs as maximum codes in polynomial metric spaces. Interactions between algebra and combinatorics. Acta Appl. Math. 29 1-2 (1992) 1-82
125. Levenshtein V.I. On designs in compact metric spaces and a universal bound on their size. Discrete Metric Spaces (Villeurbanne, 1996). Discrete Math. 192 1-3 (1998) 251-271
126. Lisoněk P. New maximal two-distance sets. J. Combin. Theory (A) 77 (1997) 318-338
127. Lyubich Y., and Vaserstein L. Isometric embeddings between classical Banach spaces, cubature formulas, and spherical designs (English summary). Geom. Dedicata 47 3 (1993) 327-362
128. Mallows C.L., Odlyzko A.M., and Sloane N.J.A. Upper bounds for modular forms, lattices, and codes. J. Algebra 36 (1975) 68-76
129. Martin W.J., Muzychuk, and Williford J.S. Imprimitive cometric association schemes: Constructions and analysis. J. Algebraic Combin. 25 (2007) 399-415
130. W.J. Martin, J.S. Williford, There are infinitely many Q-polynomial association schemes with given first multiplicity at least three, European J. Combin., in press (doi:10.1016/j.ejc.2008.07.009)
131. Mimura Y. A construction of spherical 2-design. Graphs Combin. 6 (1990) 369-372
132. Möller H.M. Lower bounds for the number of nodes in cubature formulae. Numerische Integration (Tagung, Math. Forschungsinst., Oberwolfach, 1978). Internat. Ser. Numer. Math. vol. 45 (1979), Birkhäuser, Basel, Boston, MA 221-230
133. Musin O.R. The kissing number in four dimensions. Ann. of Math. (2) 168 (2008) 1-32
134. Musin O.R. The kissing problem in three dimensions. Discrete Comput. Geom. 35 (2006) 375-384
135. O.R. Musin, Spherical two-distance sets, J. Combin. Theory (A) (in press) arXiv:0801.3706
136. Musin O.R. Multivariate positive definite functions on spheres. arXiv:math/0701083
137. Mysovskih I.P. On the evaluation of integrals over a surface of the sphere. Soviet Math. Dokl. 18 (1977) 925-929
138. Nebe G., Rains E.M., and Sloane N.J.A. Self-dual codes and invariant theory. Algorithms and Computation in Mathematics vol. 17 (2006), Springer-Verlag, Berlin xxviii+430 pp.
139. A. Neumaier, Combinatorial configurations in terms of distances (mimeographed notes), Memorandum 81-09 (Eindvohen Univ. of Technology), 1981
140. Neumaier A. Distances, graphs and designs. European J. Combin. 1 2 (1980) 163-174
141. Neumaier A., and Seidel J.J. Discrete measures for spherical designs, eutactic stars and lattices. Nederl. Akad. Wetensch. Proc. Ser. A 91 (1988) 321-334 Indag. Math. 50
142. Nikova S., and Nikov V. Improvement of the Delsarte bound for τ-designs when it is not the best bound possible. Des. Codes Cryptogr. 28 2 (2003) 201-222
143. 4. European J. Combin. 29 7 (2008) 1696-1703
144. H. Nozaki, Inside inner s-distance set and Euclidean designs, 2008, Preprint
145. H. Nozaki, M. Shinohara, On a generalization of distance sets, 2008, Preprint
146. Odlyzko A.M., and Sloane N.J.A. New bounds on the number of unit spheres that can touch a unit sphere in n dimensions. J. Combin. Theory Ser. A 26 (1979) 210-214
147. Pache C. Shells of selfdual lattices viewed as spherical designs. Internat. J. Algebra Comput. 15 5-6 (2005) 1085-1127
148. Pfender F. Improved Delsarte bounds for spherical codes in small dimensions. J. Combin. Theory Ser. A 114 6 (2007) 1133-1147
149. Pfender F., Ziegler G., and Gunter M. Kissing numbers, sphere packings, and some unexpected proofs. Notices Amer. Math. Soc. 51 8 (2004) 873-883
150. Putinar M. A note on Tchakaloff's theorem. Proc. Amer. Math. Soc. 125 8 (1997) 2409-2414
151. Rabau P., and Bajnok B. Bounds for the number of nodes in Chebyshev type quadrature formulas. J. Approx. Theory 67 (1991) 199-214
152. Reznick B. Some constructions of spherical 5-designs. Linear Algebra Appl. 226-228 (1995) 163-196
153. Reznick B. Sums of even powers of real linear forms. Mem. Amer. Math. Soc. 463 (1992)
154. Runge B. Codes and Siegel modular forms. Discrete Math. 148 1-3 (1996) 175-204
155. Saff E.B., and Kuijlaars A.B.J. Distributing many points on a sphere. Math. Intelligencer 19 (1997) 5-11
156. Sali A. On the rigidity of some spherical 2-designs. Mem. Fac. Sci. Kyushu Univ. Ser. A 47 1 (1993) 1-14
157. Sali A. On the rigidity of spherical t-designs that are orbits of finite reflection groups. Des. Codes Cryptogr. 4 2 (1994) 157-170
158. Schrijver A. New code upper bounds from the Terwilliger algebra and semidefinite programming. IEEE Trans. Inform. Theory 51 (2005) 2859-2866
159. Seidel J.J. Quasiregular two-distance sets. Nederl. Akad. Wetensch. Proc. Ser. A 72 (1969) 64-70 Indag. Math. 31
160. Seidel J.J. Spherical designs and tensors. Progress in Algebraic Combinatorics. Adv. Stud. Pure Math. vol. 24 (1996), Math. Soc., Japan, Tokyo 309-321
161. Seidel J.J. In: Corneil D.G., and Mathon (Eds). Geometry and Combinatorics, Selected Works (1991), Academic Press
162. Seki G. On some non-rigid spherical t-designs. Mem. Faculty Sci. Kyushu Univ. Ser. A Math. 46 (1992) 169-178
163. Serre J.P. Sur la lacunarite des puissances de η. Glasgow Math. J. 27 (1985) 203-221 (in French)
164. Seymour P.D., and Zaslavsky T. Averaging sets: A generalization of mean values and spherical designs. Adv. Math. 52 (1984) 213-240
165. M. Shinohara, Uniqueness of maximum three-distance sets in the three-dimensional Euclidean space, 2008, Preprint
166. Shor P.W., and Sloane N.J.A. A family of optimal packings in Grassmannian manifolds. J. Algebraic Combin. 7 (1998) 157-163
167. Shütte K., and van der Waerden B.L. Das Problem der dreizehn Kugeln. Math. Ann. 125 (1952) 325-334
168. n-dimensional Euclidean space. J. Algebraic Combin. 10 3 (1999) 279-288
169. Sidelnikov V.M. Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial. Algebra Anal. 11 4 (1999) 183-203 (in Russian). Translation in St. Petersburg Math. J. 11 (4) (2000) 673-686
170. Sloan I.H., and Womersley R.S. Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math. 21 1-2 (2004) 107-125
171. Sobolev L. Cubature formulas on the sphere invariant under finite groups of rotation. Soviet Math. Dokl. 3 (1962) 1307-1310
172. Sobolev S.L. Introduction to Theory of Cubature Formulae, Izdat (1974), Nauka, Moscow (in Russian)
173. Sobolev S.L., and Vaskevich V.L. The Theory of Cubature Formulas (1997), Kluwer Academic Publishers
174. Stroud A.H. Approximate Calculation of Multiple Integrals. Prentice-Hall Series in Automatic Computation (1971), Prentice-Hall Inc., Englewood Cliffs, NJ
175. H. Tanaka, New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra, European J. Combin., in press (doi:10.1016/j.ejc.2008.07.018)
176. Tchakaloff V. Formules de cubature mécanique à coeffcients non négatifs. Bull. Sci. Math. 81 (1857) 123-134
177. Terwilliger P. Balanced sets and Q-polynomial association schemes. Graphs Combin. 4 (1988) 87-94
178. Tiep P.H. Finite groups admitting Grassmannian 4-designs. J. Algebra 306 1 (2006) 227-243
179. Venkov B. Reseaux euclidiens, designs spheriques et formes modulaires. In: Martinet J. (Ed). Autour des travaux de Boris Venkov (On the works of Boris Venkov). Monographies de L'Enseignement Mathematique (Monographs of L'Enseignement Mathematique) vol. 37 (2001), L'Enseignement Mathematique, Geneva 272 pp (in French).
180. Venkov B. On even unimodular extremal lattice. Tr. Mat. Inst. Steklova 165 (1984) 43-48 (in Russian)
181. Venkov B. On even unimodular extremal lattice. Proc. Steklov Inst. Math. 165 3 (1985) 47-52
182. Victoir N. Asymmetric cubature formulae with few points in high dimension for symmetric measures. SIAM J. Numer. Anal. 42 (2004) 209-227
183. Wagner G. On averaging set. Monatsh. Math. 111 (1991) 69-78
184. Yudin V.A. Lower bounds for spherical designs. Izv. Ross. Akad. Nauk. Ser. Mat. 61 3 (1997) 211-233 English transl., Izv. Math. 61 (1997) 673-683
185. Yudin V.A. Distribution of the points of a design on a sphere. Izv. Ross. Akad. Nauk Ser. Mat. 69 5 (2005) 205-224 (in Russian). Translation in Izv. Math. 69 (5) (2005) 1061-1079