球形码的泛界：Levenshtein框架提升
Universal bounds for spherical codes: the Levenshtein framework lifted
摘要：
基于Delsarte-Yudin线性规划方法，我们推广Levenshtein的框架，得到了具有指定维数和基数的球码的最小$h$ -能量的下界，以及具有指定维数和最小间隔的球码的最大基数的上界.这些界限是普遍的意义上说，他们持有的一大类潜在的$h$和Levenshtein的意义。此外，达到上界的码在Cohn-Kumar意义下是普遍最优的。把Levenshtein界和能量界看作"第一级”，我们的结果可以看作是“下一级”泛界，因为它们具有相同的一般性质，并隐含了局部和全局最优性的充要条件.为此，我们引入了泛下界空间（ULB空间）的概念，一个空间，满足一定的正交和插值性质。虽然我们的方法适用于许多情况，但我们将强调$\mathbb{S}^3$上的$24$点（$24$ -cell）和$120$点（$600$ -cell）的模型示例。特别是，我们提供了一个新的证明，$600$ -细胞是普遍最优的，并在这样做，我们得到的最优性的$600$ -细胞一类大于绝对单调的潜力考虑科恩-库马尔。
英文摘要：
Based on the Delsarte-Yudin linear programming approach, we extend Levenshtein's framework to obtain lower bounds for the minimum $h$-energy of spherical codes of prescribed dimension and cardinality, and upper bounds on the maximal cardinality of spherical codes of prescribed dimension and minimum separation. These bounds are universal in the sense that they hold for a large class of potentials $h$ and in the sense of Levenshtein. Moreover, codes attaining the bounds are universally optimal in the sense of Cohn-Kumar. Referring to Levenshtein bounds and the energy bounds of the authors as ``first level", our results can be considered as ``next level" universal bounds as they have the same general nature and imply necessary and sufficient conditions for their local and global optimality. For this purpose, we introduce the notion of Universal Lower Bound space (ULB-space), a space that satisfies certain quadrature and interpolation properties. While there are numerous cases for which our method applies, we will emphasize the model examples of $24$ points ($24$-cell) and $120$ points ($600$-cell) on $\mathbb{S}^3$. In particular, we provide a new proof that the $600$-cell is universally optimal, and in so doing, we derive optimality of the $600$-cell on a class larger than the absolutely monotone potentials considered by Cohn-Kumar.
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