论文标题
球形代码和包装的新上限
New upper bounds for spherical codes and packings
论文作者
论文摘要
我们在每$θ<θ^*\每一个$θ$ spherical代码上提高了以前最著名的上限，每$θ<θ^*\大约62.997^{\ circ} $，至少在足够高的尺寸上以$ 0.4325 $的倍数。此外，对于尺寸的球体包装密度$ n \ geq 2000 $，我们至少有一个0.4325+\ frac {51} {n} $的改进。我们的方法还打破了许多非数字球体堆积密度界限，尺寸较小。这是每个维度的第一个改进，因为Kabatyanskii和Levenshtein〜 \ cite {kl}的工作，levenshtein〜 \ cite \ cite {leven79}的后来改进。本文的新颖性包括分析三个相关性，高度质量浓度的使用以及对雅各比多项式根部之间间距的研究。
We improve the previously best known upper bounds on the sizes of $θ$-spherical codes for every $θ<θ^*\approx 62.997^{\circ}$ 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.
