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$O(n \log n)$.

We present a new approach to construct spherical codes in 2kdimensions, based on Hopf foliations. Using the fact that a sphere S2n-1is foliated by manifolds Scos ηn-1× Ssin ηn-1, η ∈ [0, π/2], we distribute points in dimension 2kvia a recursive algorithm from a basic construction in R4. Our procedure outperforms some current constructive methods in several small-distance regimes and constitutes a compromise between optimality and computational effort.