New upper bounds for spherical codes and packings

TL;DR

This paper presents new upper bounds for spherical codes and sphere packings in high dimensions, significantly improving previous bounds by analyzing triple correlations, high-dimensional mass concentration, and Jacobi polynomial roots.

Contribution

It introduces novel methods including triple correlation analysis and root spacing studies to improve bounds on spherical codes and packings across dimensions.

Findings

Improved upper bounds for spherical codes below 63° in high dimensions.

Enhanced sphere packing density bounds for dimensions ≥2000.

First improvements since classical bounds by Kabatyanskii and Levenshtein.

Abstract

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.