Constructive Spherical Codes by Hopf Foliations

TL;DR

This paper introduces a recursive method for constructing spherical codes in dimensions that are powers of two, leveraging Hopf foliations to improve code density and computational efficiency, with practical encoding and decoding algorithms.

Contribution

It presents a novel systematic approach to spherical code construction using Hopf foliations, outperforming existing methods in certain regimes and offering efficient encoding and decoding procedures.

Findings

Outperforms some existing methods in small-distance regimes

Provides bounds for asymptotic density of codes

Offers low-complexity encoding and decoding algorithms

Abstract

We present a new systematic approach to constructing spherical codes in dimensions $2^k$, based on Hopf foliations. Using the fact that a sphere $S^{2n-1}$ is foliated by manifolds $S_{\cos\eta}^{n-1} \times S_{\sin\eta}^{n-1}$, $\eta\in[0,\pi/2]$, we distribute points in dimension $2^k$ via a recursive algorithm from a basic construction in $\mathbb{R}^4$. Our procedure outperforms some current constructive methods in several small-distance regimes and constitutes a compromise between achieving a large number of codewords for a minimum given distance and effective constructiveness with low encoding computational cost. Bounds for the asymptotic density are derived and compared with other constructions. The encoding process has storage complexity $O(n)$ and time complexity $O(n \log n)$. We also propose a sub-optimal decoding procedure, which does not require storing the codebook and has time complexity $O(n \log n)$.