## 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.

Original language | English |
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Pages (from-to) | 609-625 |

Number of pages | 17 |

Journal | St. Petersburg Mathematical Journal |

Volume | 16 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2005 |

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Applied Mathematics