TY - JOUR

T1 - On spherical designs of some harmonic indices

AU - Zhu, Yan

AU - Bannai, Eiichi

AU - Bannai, Etsuko

AU - Kim, Kyoung Tark

AU - Yu, Wei Hsuan

N1 - Publisher Copyright:
© 2017, Australian National University. All rights reserved.

PY - 2017/4/13

Y1 - 2017/4/13

N2 - 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: Σx∈Yf(x) = 0 for all real homogeneous harmonic polynomials f(x1, . . . , xn) 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 Σx∈Yf(x) = 0 is satisfied for all real homogeneous harmonic polynomials f(x1, . . . , xn) 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}.

AB - 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: Σx∈Yf(x) = 0 for all real homogeneous harmonic polynomials f(x1, . . . , xn) 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 Σx∈Yf(x) = 0 is satisfied for all real homogeneous harmonic polynomials f(x1, . . . , xn) 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}.

KW - Delsarte’s method

KW - Elliptic diophantine equation

KW - Fisher type lower bound

KW - Gegenbauer polynomial

KW - Larman-Rogers-Seidel’s theorem

KW - Semidefinite programming

KW - Spherical designs of harmonic index

KW - Tight design

UR - http://www.scopus.com/inward/record.url?scp=85018508280&partnerID=8YFLogxK

U2 - 10.37236/6437

DO - 10.37236/6437

M3 - 期刊論文

AN - SCOPUS:85018508280

SN - 1077-8926

VL - 24

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 2

ER -