TY - JOUR

T1 - Bounds on antipodal spherical designs with few angles

AU - Xu, Zhiqiang

AU - Xu, Zili

AU - Yu, Wei Hsuan

N1 - Publisher Copyright:
© The authors.

PY - 2021

Y1 - 2021

N2 - A finite subset X on the unit sphere Sd is called an s-distance set with strength t if its angle set A(X):= {〈x, y〉: x, y ∈ X, x ≠ y} has size s, and X is a spherical t-design but not a spherical (t + 1)-design. In this paper, we consider to estimate the maximum size of such antipodal set X for small s. Motivated by the method developed by Nozaki and Suda, for each even integer s ∈ [t+5 2, t + 1] with t ≥ 3, we improve the best known upper bound of Delsarte, Goethals and Seidel. We next focus on two special cases: s = 3, t = 3 and s = 4, t = 5. Estimating the size of X for these two cases is equivalent to estimating the size of real equiangular tight frames (ETFs) and Levenstein-equality packings, respectively. We improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives an upper bound on |X| when s = 3, t = 3 and s = 4, t = 5, respectively.

AB - A finite subset X on the unit sphere Sd is called an s-distance set with strength t if its angle set A(X):= {〈x, y〉: x, y ∈ X, x ≠ y} has size s, and X is a spherical t-design but not a spherical (t + 1)-design. In this paper, we consider to estimate the maximum size of such antipodal set X for small s. Motivated by the method developed by Nozaki and Suda, for each even integer s ∈ [t+5 2, t + 1] with t ≥ 3, we improve the best known upper bound of Delsarte, Goethals and Seidel. We next focus on two special cases: s = 3, t = 3 and s = 4, t = 5. Estimating the size of X for these two cases is equivalent to estimating the size of real equiangular tight frames (ETFs) and Levenstein-equality packings, respectively. We improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives an upper bound on |X| when s = 3, t = 3 and s = 4, t = 5, respectively.

UR - http://www.scopus.com/inward/record.url?scp=85112266144&partnerID=8YFLogxK

U2 - 10.37236/9891

DO - 10.37236/9891

M3 - 期刊論文

AN - SCOPUS:85112266144

VL - 28

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 3

M1 - P3.39

ER -