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The nonexistence of certain tight spherical designs


Authors: E. Bannai, A. Munemasa and B. Venkov
Original publication: Algebra i Analiz, tom 16 (2004), nomer 4.
Journal: St. Petersburg Math. J. 16 (2005), 609-625
MSC (2000): Primary 05B30
DOI: https://doi.org/10.1090/S1061-0022-05-00868-X
Published electronically: June 21, 2005
MathSciNet review: 2090848
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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.


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Additional Information

E. Bannai
Affiliation: Graduate School of Mathematics, Kyushu University, Hakozaki 6-10-1, Higashi-ku, Fukuoka 812-8581, Japan
Email: bannai@math.kyushu-u.ac.jp

A. Munemasa
Affiliation: Graduate School of Information Sciences, Tohoku University, Aramaki-Aza-Aoba 09 Aoba-ku, Sendai 980-8579, Japan
Email: munemasa@math.is.tohoku.ac.jp

B. Venkov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: bbvenkov@yahoo.com

DOI: https://doi.org/10.1090/S1061-0022-05-00868-X
Received by editor(s): September 3, 2003
Published electronically: June 21, 2005
Additional Notes: The third author was partially supported by the Swiss National Science Foundation. This work was done during a three month visit of B. Venkov to Kyushu University, and he thanks the University for hospitality.
The paper contains Appendix written by Y.-F. S. Pétermann.
Article copyright: © Copyright 2005 American Mathematical Society

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