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Constructive Spherical Codes by Hopf Foliations.

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Bibliographic Details
Title: Constructive Spherical Codes by Hopf Foliations.
Authors: Miyamoto, Henrique K., Costa, Sueli I. R., Earp, Henrique N. Sa
Source: IEEE Transactions on Information Theory; Dec2021, Vol. 67 Issue 12, p7925-7939, 15p
Subject Terms: FOLIATIONS (Mathematics), CHANNEL coding, EUCLIDEAN distance
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 $O(n \log n)$. [ABSTRACT FROM AUTHOR]
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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 $O(n \log n)$. [ABSTRACT FROM AUTHOR]
ISSN:00189448
DOI:10.1109/TIT.2021.3114094