The nonexistence of certain tight spherical designs. (English) Zbl 1072.05017
St. Petersbg. Math. J. 16, No. 4, 609-625 (2005) and Algebra Anal. 16, No. 4, 1-23 (2004).
Summary: The nonexistence of tight spherical designs is shown in some cases left open to date. Tight spherical 5-designs may exist in dimension \(n=(2m+1)^2-2\) and the existence is known only for \(m=1, 2\). In the paper, the existence is ruled out under a certain arithmetic condition on the integer \(m\), satisfied by infinitely many values of \(m\) including \(m=4\). Also, nonexistence is shown for \(m=3\). Tight spherical 7-designs may exist in dimension \(n=3d^2-4\), and the existence is known only for \(d=2,3\). In the paper, the existence is ruled out under a certain arithmetic condition on \(d\), satisfied by infinitely many values of \(d\), including \(d=4\). Also, nonexistence is shown for \(d=5\). The fact that the arithmetic conditions on \(m\) for 5-designs and on \(d\) for 7-designs are satisfied by infinitely many values of \(m\) and \(d\), respectively, is shown in the Appendix written by Y.-F. S. Pétermann.
MSC:
| 05B30 | Other designs, configurations |
References:
| [1] | E. Bannai and E. Bannai, Algebraic combinatorics on spheres, Springer-Verlag, Tokyo, 1999. (Japanese) · Zbl 1207.05022 |
| [2] | E. Bannai and R. M. Damerell, Tight spherical designs. I, J. Math. Soc. Japan 31 (1979), no. 1, 199 – 207. · Zbl 0403.05022 · doi:10.2969/jmsj/03110199 |
| [3] | E. Bannai and R. M. Damerell, Tight spherical designs. II, J. London Math. Soc. (2) 21 (1980), no. 1, 13 – 30. · Zbl 0436.05018 · doi:10.1112/jlms/s2-21.1.13 |
| [4] | Eiichi Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canad. J. Math. 33 (1981), no. 2, 437 – 449. · doi:10.4153/CJM-1981-038-7 |
| [5] | L. Carlitz, On a problem of additive arithmetic. II, Quart. J. Math. Oxford Ser. 3 (1932), 273-290. · Zbl 0006.10401 |
| [6] | J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1999. 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. · Zbl 0915.52003 |
| [7] | P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), no. 3, 363 – 388. · Zbl 0376.05015 |
| [8] | J.-M. Goethals and J. J. Seidel, Spherical designs, Relations between combinatorics and other parts of mathematics (Proc. Sympos. Pure Math., Ohio State Univ., Columbus, Ohio, 1978) Proc. Sympos. Pure Math., XXXIV, Amer. Math. Soc., Providence, R.I., 1979, pp. 255 – 272. · Zbl 0404.05015 |
| [9] | J.-M. Goethals and J. J. Seidel, The regular two-graph on 276 vertices, Discrete Math. 12 (1975), 143 – 158. · Zbl 0311.05121 · doi:10.1016/0012-365X(75)90029-1 |
| [10] | D. R. Heath-Brown, The square sieve and consecutive square-free numbers, Math. Ann. 266 (1984), no. 3, 251 – 259. · Zbl 0514.10038 · doi:10.1007/BF01475576 |
| [11] | Jacques Martinet, Sur certains designs sphériques liés à des réseaux entiers, Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math., vol. 37, Enseignement Math., Geneva, 2001, pp. 135 – 146 (French, with English and French summaries). · Zbl 1065.11050 |
| [12] | John Milnor and Dale Husemoller, Symmetric bilinear forms, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. · Zbl 0292.10016 |
| [13] | L. Mirsky, On the frequency of pairs of square-free numbers with a given difference, Bull. Amer. Math. Soc. 55 (1949), 936 – 939. · Zbl 0035.31301 |
| [14] | L. Mirsky, Arithmetical pattern problems relating to divisibility by \?th powers, Proc. London Math. Soc. (2) 50 (1949), 497 – 508. · Zbl 0031.25209 · doi:10.1112/plms/s2-50.7.497 |
| [15] | Gabriele Nebe and Boris Venkov, The strongly perfect lattices of dimension 10, J. Théor. Nombres Bordeaux 12 (2000), no. 2, 503 – 518 (English, with English and French summaries). Colloque International de Théorie des Nombres (Talence, 1999). · Zbl 0997.11049 |
| [16] | Boris Venkov, Réseaux et designs sphériques, Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math., vol. 37, Enseignement Math., Geneva, 2001, pp. 10 – 86 (French, with English and French summaries). · Zbl 1139.11320 |
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