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数下对于每个$\theta<\theta^*\approx 62.997^{\circ}$的$\theta$-球面码大小的先前已知最优上界，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。我们的方法还打破了小维度中的许多球体填充密度界限。 Apart from Cohn and Zhao's~\cite{CohnZhao} improvement on the geometric average of Levenshtein's bound~\cite{Leven79} over all sufficiently high dimensions by a factor of $0.79,$ our work is the first improvement for each dimension since the work of Kabatyanskii and Levenshtein~\cite{KL} and its later improvement by Levenshtein~\cite{Leven79}. Moreover, we generalize Levenshtein's optimal polynomials and provide explicit formulae for them that may be of independent interest. 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 sphere packing density bounds in small dimensions. Apart from Cohn and Zhao's~\cite{CohnZhao} improvement on the geometric average of Levenshtein's bound~\cite{Leven79} over all sufficiently high dimensions by a factor of $0.79,$ our work is the first improvement for each dimension since the work of Kabatyanskii and Levenshtein~\cite{KL} and its later improvement by Levenshtein~\cite{Leven79}. Moreover, we generalize Levenshtein's optimal polynomials and provide explicit formulae for them that may be of independent interest.