基于Hopf叶的构造性球面码
Constructive Spherical Codes by Hopf Foliations
摘要：
我们提出了一个新的系统的方法来构建球形码的尺寸$2^k$，基于霍普夫叶。利用球面$S^{2n-1}$由流形$S_{\cosη}^{n-1} \times S_{\sinη}^{n-1}$，$η\in[0,π/2]$构成的事实，从$\mathbb{R}^4$的基本构造出发，通过递归算法在$2^k$维上分布点.我们的程序优于目前的一些建设性的方法，在几个小距离制度，并构成了一个妥协，实现了大量的码字的最小给定的距离和有效的建设性与低编码计算成本。渐近密度的界限推导和比较与其他建设。编码过程具有存储复杂度$O(n)$和时间复杂度$O(n \log n)$。我们还提出了一个次优的解码过程，它不需要存储的码本，并具有时间复杂度$O(n \log n)$。
英文摘要：
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 $S_{\cosη}^{n-1} \times S_{\sinη}^{n-1}$, $η\in[0,π/2]$, 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 $O(n \log n)$.
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