Stiefel manifolds and upper bounds for spherical codes and packings

Abstract

We improve upper bounds on sphere packing densities and sizes of spherical codes in high dimensions. In particular, we prove that the maximal sphere packing densities $\delta_n$ in $\mathbb{R}^n$ satisfy \[\delta_n\leq \frac{1+o(1)}{e}\cdot \delta^{\text{KL}}_{n}\] for large $n$, where $\delta^{\text{KL}}_{n}$ is the best bound on $\delta_n$ obtained essentially by Kabatyanskii and Levenshtein from the 1970s with improvements over the years. We also obtain the same improvement factor for the maximal size $M(n,\theta)$ of $\theta$-spherical codes in $S^{n-1}$: for angles $0<\theta<\theta'\leq\frac{\pi}{2}$, \[M(n,\theta)\leq \frac{1+o(1)}{e}\cdot \frac{M_{\text{Lev}}(n-1,\theta')}{\mu_n(\theta,\theta')}\] for large $n$, where $\mu_n(\theta,\theta')$ is the mass of the spherical cap in the unit sphere $S^{n-1}$ of radius $\frac{\sin(\theta/2)}{\sin(\theta'/2)}$, and $M_{\text{Lev}}(n-1,\theta')$ is Levenshtein's upper bound on $M(n-1,\theta')$ when applying the Delsarte linear programming method to Levenshtein's optimal polynomials. In fact, we prove that there are no analytic losses in our arguments and that the constant $\frac{1}{e}=0.367...$ is optimal for the class of functions considered. Our results also show that the improvement factor does not depend on the special angle $\theta^*=62.997...^{\circ}$, explaining the numerics in arXiv:2001.00185. In the spherical codes case, the above inequality improves the Kabatyanskii--Levenshtein bound by a factor of $0.2304...$ on geometric average. Along the way, we construct a general class of functions using Stiefel manifolds for which we prove general results and study the improvement factors obtained from them in various settings.and study the improvement factors obtained from them in various settings.