Bounds on antipodal spherical designs with few angles

2020/07/28 by Zhiqiang Xu, Xu, Zhiqiang, Zili Xu +3 · 1 citation
Mathematics · Computer Science · Materials Science · #Mathematical Approximation and Integration #Digital Image Processing Techniques #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2007.13999

Abstract

A finite subset X on the unit sphere \mathbbS^d-1 is called an s-distance set with strength t if its angle set A(X):=\⟨ x,y⟩ : x,y∈ X,x\neqy \ 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 for small s. First, we improve the known bound on |X| for each even integer s∈[(t+5)/(2), t+1] when t≥ 3. 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 first improve the previous estimate on the size of real ETFs and Levenstein-equality packings. This in turn gives a bound on |X| when s=3, t=3 and s=4, t=5, respectively.

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