Constructive Spherical Codes by Hopf Foliations

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 $O(n\log n)$.

15 pages, 9 figures, minor improvements. Accepted to the IEEE Transactions on Information Theory

References in corpus (1)

• Phase Shift Keying on the Hypersphere: Peak Power-Efficient MIMO Communications