## Abstract

A finite subset Y on the unit sphere S^{n−1}⊆ ℝ n is called a spherical design of harmonic index t, if the following condition is satisfied: Σ_{x}_{∈Y}f(x) = 0 for all real homogeneous harmonic polynomials f(x_{1}, . . . , x_{n}) of degree t. Also, for a subset T of ℕ = {1, 2, · · · }, a finite subset Y ⊆ S^{n−1}is called a spherical design of harmonic index T, if Σ_{x}_{∈Y}f(x) = 0 is satisfied for all real homogeneous harmonic polynomials f(x_{1}, . . . , x_{n}) of degree k with k ∈ T. In the present paper we first study Fisher type lower bounds for the sizes of spherical designs of harmonic index t (or for harmonic index T). We also study ‘tight’ spherical designs of harmonic index t or index T. Here ‘tight’ means that the size of Y attains the lower bound for this Fisher type inequality. The classification problem of tight spherical designs of harmonic index t was started by Bannai-OkudaTagami (2015), and the case t = 4 was completed by Okuda-Yu (2016). In this paper we show the classification (non-existence) of tight spherical designs of harmonic index 6 and 8, as well as the asymptotic non-existence of tight spherical designs of harmonic index 2e for general e ≥ 3. We also study the existence problem for tight spherical designs of harmonic index T for some T, in particular, including index T = {8, 4}.

Original language | English |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 24 |

Issue number | 2 |

DOIs | |

State | Published - 13 Apr 2017 |

## Keywords

- Delsarte’s method
- Elliptic diophantine equation
- Fisher type lower bound
- Gegenbauer polynomial
- Larman-Rogers-Seidel’s theorem
- Semidefinite programming
- Spherical designs of harmonic index
- Tight design