Stiefel manifolds and upper bounds for spherical codes and packings
Masoud Zargar
TL;DR
The paper advances long-standing high-dimensional sphere-packing and spherical-code bounds by developing a unified averaging framework that preserves positive definiteness and LP-compatibility. By constructing general functions via averaging over projections, including over Stiefel manifolds, and leveraging sharp Jacobi/Krasikov polynomial estimates, the authors establish an angle-independent improvement factor of 1/e over the Kabatiansky–Levenshtein and Levenshtein bounds, respectively, for large dimensions. They prove δ_n ≤ (1+o(1))/e · δ_n^{KL} and M(n,θ) ≤ (1+o(1))/e · M_{Lev}(n-1,θ')/μ_n(θ,θ') and show optimality within the considered function class; they also quantify improvement factors for various m (notably m=1,2, and at least some m≥3) and several regimes, including n−m constant. The work connects averaging over Stiefel manifolds to improved bounds, offering new geometric and analytic tools that sharpen classical LP-type bounds in coding and packing theory. Overall, the results provide deeper insight into how higher-dimensional projections and refined polynomial estimates yield concrete tightening of fundamental packing and coding limits in high dimensions.
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 $δ_n$ in $\mathbb{R}^n$ satisfy \[δ_n\leq \frac{1+o(1)}{e}\cdot δ^{\text{KL}}_{n}\] for large $n$, where $δ^{\text{KL}}_{n}$ is the best bound on $δ_n$ 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 $M(n,θ)$ of $θ$-spherical codes in $S^{n-1}$: for angles $0<θ<θ'\leq\fracπ{2}$, \[M(n,θ)\leq \frac{1+o(1)}{e}\cdot \frac{M_{\text{Lev}}(n-1,θ')}{μ_n(θ,θ')}\] for large $n$, where $μ_n(θ,θ')$ is the mass of the spherical cap in the unit sphere $S^{n-1}$ of radius $\frac{\sin(θ/2)}{\sin(θ'/2)}$, and $M_{\text{Lev}}(n-1,θ')$ is Levenshtein's upper bound on $M(n-1,θ')$ 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 $θ^*=62.997...^{\circ}$, 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.