The nonexistence of certain tight spherical designs

Bannai, E.; Munemasa, A.; Venkov, B.

doi:10.1090/S1061-0022-05-00868-X

The nonexistence of certain tight spherical designs
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by E. Bannai, A. Munemasa and B. Venkov

St. Petersburg Math. J. 16 (2005), 609-625

DOI: https://doi.org/10.1090/S1061-0022-05-00868-X

Published electronically: June 21, 2005

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Abstract:

In this paper, 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.

References

E. Bannai and E. Bannai, Algebraic combinatorics on spheres, Springer-Verlag, Tokyo, 1999. (Japanese)
E. Bannai and R. M. Damerell, Tight spherical designs. I, J. Math. Soc. Japan 31 (1979), no. 1, 199–207. MR 519045, DOI 10.2969/jmsj/03110199
E. Bannai and R. M. Damerell, Tight spherical designs. II, J. London Math. Soc. (2) 21 (1980), no. 1, 13–30. MR 576179, DOI 10.1112/jlms/s2-21.1.13
Eiichi Bannai and N. J. A. Sloane, Uniqueness of certain spherical codes, Canadian J. Math. 33 (1981), no. 2, 437–449. MR 617634, DOI 10.4153/CJM-1981-038-7
L. Carlitz, On a problem of additive arithmetic. II, Quart. J. Math. Oxford Ser. 3 (1932), 273–290.
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. MR 1662447, DOI 10.1007/978-1-4757-6568-7
P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), no. 3, 363–388. MR 485471, DOI 10.1007/bf03187604
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. MR 525330
J.-M. Goethals and J. J. Seidel, The regular two-graph on $276$ vertices, Discrete Math. 12 (1975), 143–158. MR 384597, DOI 10.1016/0012-365X(75)90029-1
D. R. Heath-Brown, The square sieve and consecutive square-free numbers, Math. Ann. 266 (1984), no. 3, 251–259. MR 730168, DOI 10.1007/BF01475576
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). MR 1878748
John Milnor and Dale Husemoller, Symmetric bilinear forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73, Springer-Verlag, New York-Heidelberg, 1973. MR 0506372
L. Mirsky, On the frequency of pairs of square-free numbers with a given difference, Bull. Amer. Math. Soc. 55 (1949), 936–939. MR 31507, DOI 10.1090/S0002-9904-1949-09313-8
L. Mirsky, Arithmetical pattern problems relating to divisibility by $r$th powers, Proc. London Math. Soc. (2) 50 (1949), 497–508. MR 28334, DOI 10.1112/plms/s2-50.7.497
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). MR 1823200
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). MR 1878745