通过字母表扩展加速极化
Accelerating Polarization via Alphabet Extension
AI总结：
本文提出四面体删除信道并采用 GF(4) 上的 Mori–Tanaka 2×2 矩阵构造极化码，证明其缩放指数上界为 3.328，表明扩展字母表比增大矩阵更能有效加速极化。
摘要：
极化是一种前所未有的编码技术，因为它不仅能够达到信道容量，而且其收敛速度比任何其他编码技术都更快。这一速度由“缩放指数”衡量，其重要性体现在三个方面。首先，估计缩放指数具有挑战性，需要更深入地理解通信信道的动力学。其次，缩放指数可作为不同极化码变体的基准，帮助我们为实际应用选择合适的变体。第三，优化缩放指数的需求为如何强化极化码设计提供了启示。在本文中，我们将二元删除信道（BEC）这一最简单且在许多编码理论研究中占据核心地位的通信信道，推广为“四面体删除信道”（TEC）。随后，我们调用 Mori–Tanaka 在 GF(4) 上的 2×2 矩阵，在 TEC 上构造极化码。我们的主要贡献是证明 TEC 的动力学收敛到一个几乎单参数的信道族，进而得到缩放指数的上界 3.328。这是首个缩放指数被给出上界的非二元矩阵。它对 BEC 的极化速度也快于所有已知的、尺寸最大至 23×23 的二元矩阵。我们的结果表明，为实现更快的极化，扩展字母表是增大矩阵尺寸之外一种更有效、更实用的替代方案。
英文摘要：
Polarization is an unprecedented coding technique in that it not only achieves channel capacity, but also does so at a faster speed of convergence than any other coding technique. This speed is measured by the ``scaling exponent'' and its importance is three-fold. Firstly, estimating the scaling exponent is challenging and demands a deeper understanding of the dynamics of communication channels. Secondly, scaling exponents serve as a benchmark for different variants of polar codes that helps us select the proper variant for real-life applications. Thirdly, the need to optimize for the scaling exponent sheds light on how to reinforce the design of polar codes. In this paper, we generalize the binary erasure channel (BEC), the simplest communication channel and the protagonist of many coding theory studies, to the ``tetrahedral erasure channel'' (TEC). We then invoke Mori--Tanaka's $2 \times 2$ matrix over GF$(4)$ to construct polar codes over TEC. Our main contribution is showing that the dynamic of TECs converges to an almost--one-parameter family of channels, which then leads to an upper bound of $3.328$ on the scaling exponent. This is the first non-binary matrix whose scaling exponent is upper-bounded. It also polarizes BEC faster than all known binary matrices up to $23 \times 23$ in size. Our result indicates that expanding the alphabet is a more effective and practical alternative to enlarging the matrix in order to achieve faster polarization.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
