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.

翻译：我们改进了先前已知的关于$\theta$-球码大小的上界，对于每个$\theta<\theta^*\approx 62.997^{\circ}$，在足够高的维度上至少提高了0.4325倍。此外，对于维度$n\geq 2000$的球堆积密度，我们至少提高了$0.4325+\frac{51}{n}$倍。我们的方法还打破了较小维度中许多非数值的球堆积密度上界。这是自Kabatyanskii和Levenshtein~\cite{KL}的工作及其后来由Levenshtein~\cite{Leven79}改进以来，每个维度的首次此类改进。本文的新颖之处包括三重相关性分析、高维质量集中性的应用以及雅可比多项式根之间间距的研究。