标题： 球面码和包装的新上界 显示英文标题

标题： New upper bounds for spherical codes and packings

摘要： 我们改进了在足够高维情况下，对于每个$\theta<\theta^*\approx 62.997^{\circ}$的$\theta$-球面码大小的先前已知上界，至少提高了$0.4325$倍。 此外，对于维度$n\geq 2000$的球体填充密度，我们至少提高了$0.4325+\frac{51}{n}$倍。 我们的方法还在较小维度中打破了诸多非数值的球体填充密度界限。 这是自 Kabatyanskii 和 Levenshtein~\cite{KL}的工作及其后来由 Levenshtein~\cite{Leven79}的改进以来，每个维度上的首次此类改进。 本文的创新之处包括对三重相关性的分析、高维质量集中现象的使用，以及对 Jacobi 多项式根之间间距的研究。 显示英文摘要

摘要： We improve the previously best known upper bounds on the sizes of $\theta$-spherical codes for every $\theta<\theta^*\approx 62.997^{\circ}$ at least by a factor of $0.4325$, in sufficiently high dimensions. Furthermore, for sphere packing densities in dimensions $n\geq 2000$ we have an improvement at least by a factor of $0.4325+\frac{51}{n}$. Our method also breaks many non-numerical sphere packing density bounds in smaller dimensions. This is the first such improvement for each dimension since the work of Kabatyanskii and Levenshtein~\cite{KL} and its later improvement by Levenshtein~\cite{Leven79}. Novelties of this paper include the analysis of triple correlations, usage of the concentration of mass in high dimensions, and the study of the spacings between the roots of Jacobi polynomials.