### Abstract

Spherical designs were introduced by Delsarte, Goethals, and Seidel in 1977. A spherical t-design in, R^{n} is a finite set X ⊂ S^{n-1} with the property that for every polynomial p with degree ≤ t, the average value of p on X equals the average value of p on S^{n-1}. This paper contains some existence and nonexistence results, mainly for spherical 5-designs in R^{3}. Delsarte, Goethals, and Seidel proved that if X is a spherical 5-design in R^{3}, then |X| ≥ 12 and if |Xz.sfnc; = 12, then X consists of the vertices of a regular icosahedron. We show that such designs exist with cardinality 16, 18, 20, 22, 24, and every integer ≥ 26. If X is a spherical 5-design in R^{n}, then |X| ≥ n(n + 1); if |X| = n(n + 1), then X has been called tight. Tight spherical 5-designs in R^{n} are known to exist only for n = 2, 3, 7, 23 and possibly n = u^{2} - 2 for odd u ≥ 7. Any tight spherical 5-design in R^{n} must consist of n(n + 1) 2 antipodal pairs of points. We show that for n ≥ 3, there are no spherical 5-designs in R^{n} consisting of n(n + 1) 2 + 1 antipodal pairs of points.

Original language | English (US) |
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Pages (from-to) | 163-196 |

Number of pages | 34 |

Journal | Linear Algebra and Its Applications |

Volume | 226-228 |

Issue number | C |

DOIs | |

State | Published - 1995 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

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## Cite this

*Linear Algebra and Its Applications*,

*226-228*(C), 163-196. https://doi.org/10.1016/0024-3795(95)00101-V