Constructive Spherical Codes by Hopf Foliations
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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_cosη^n-1× S_sinη^n-1, η∈[0,π/2], we distribute points in dimension 2^k via a recursive algorithm from a basic construction in 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).
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