论文标题
在抗虫球形设计上的边界很少
Bounds on antipodal spherical designs with few angles
论文作者
论文摘要
单位球上的有限子集$ x $ $ \ mathbb {s}^{d-1} $称为$ s $ distance套装，如果其角度集$ a（x）$ a（x）：= \ {\ langle \ mathbf {x} \ Mathbf {X}，\ Mathbf {Y} \ in X，\ Mathbf {X} \ Neq \ MathBf {Y} \} $具有size $ s $，而$ x $是球形$ t $ - deSign，但不是球形$（t+1）$ - 设计。在本文中，我们考虑估计小额$ S $的这种反物设置的最大尺寸。首先，我们在[\ frac {t+5} {2} {2} {2}，t+1] $时，在$ | x | $上改进了已知的绑定，当$ t \ geq 3 $。我们接下来要关注两种特殊情况：$ s = 3，\ t = 3 $和$ s = 4，\ t = 5 $。这两种情况的估计$ x $的大小等同于估计分别估计真实等缘紧密框架（ETF）和莱文斯坦 - 平等包装的大小。我们首先改善了对实际ETF和Levenstein-Equality包装大小的估计。反过来，当$ s = 3，\ t = 3 $和$ s = 4，\ t = 5 $时，这给出了$ | x | $。
A finite subset $X$ on the unit sphere $\mathbb{S}^{d-1}$ is called an $s$-distance set with strength $t$ if its angle set $A(X):=\{\langle \mathbf{x},\mathbf{y}\rangle : \mathbf{x},\mathbf{y}\in X,\mathbf{x}\neq\mathbf{y} \}$ 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 for small $s$. First, we improve the known bound on $|X|$ for each even integer $s\in[\frac{t+5}{2}, t+1]$ when $t\geq 3$. 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.
