论文标题
HOPF叶子的建设性球形代码
Constructive Spherical Codes by Hopf Foliations
论文作者
论文摘要
我们提出了一种基于HOPF叶子的$ 2^k $构建球形代码的新系统方法。 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 $。我们的程序在几个小距离制度中的当前建设性方法优于某些当前的建设性方法，并且构成了以最小给定距离获得大量代码字之间的妥协，并且具有低编码的计算成本，有效的建设性。得出了渐近密度的边界，并将其与其他结构进行比较。编码过程具有存储复杂性$ O（n）$和时间复杂性$ O（n \ log n）$。我们还提出了一个次优的解码过程，该过程不需要存储代码簿，并且具有时间复杂性$ o（n \ log n）$。
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 $O(n \log n)$.
