On spherical designs of some harmonic indices

A finite subset Y on the unit sphere Sⁿ⁻¹⊆ ℝ n is called a spherical design of harmonic index t, if the following condition is satisfied: Σ_x∈Yf(x) = 0 for all real homogeneous harmonic polynomials f(x₁, . . . , x_n) of degree t. Also, for a subset T of ℕ = {1, 2, · · · }, a finite subset Y ⊆ Sⁿ⁻¹is called a spherical design of harmonic index T, if Σ_x∈Yf(x) = 0 is satisfied for all real homogeneous harmonic polynomials f(x₁, . . . , 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}.