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 $O(n \log n)$.

翻译：我们提出一个新的系统方法,根据霍普夫版图,在2美元的范围内构建球形代码。使用一个球体$S ⁇ 2n-1美元是由数元($S ⁇ cos\eta ⁇ n-1}\ times S ⁇ sin\eta ⁇ n-1}$\eta\in[0,\pi/2]美元,我们提出一个新的系统方法,在2美元的范围内构建球形代码。我们的程序在几个小距离制度中超越了目前一些建设性的方法,并且构成了一种妥协,即为了一个最小的定线距离和低编码成本的有效建设性性而实现大量编码词。对无源密度的计算是衍生出来的,与其他构造进行比较。编码过程具有存储复杂性$(n)美元和时间复杂性$O(n\log n)美元。我们还提议了一个亚最佳解码程序,不需要存储代码簿,而有时间复杂性$O(n\log n)美元。