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.

翻译：极化是一种前所未有的编码技术，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更快。这种速度通过“标度指数”来衡量，其重要性体现在三个方面。首先，估计算法标度指数具有挑战性，需要更深入理解通信信道的动态特性。其次，标度指数作为不同极化码变体的基准，有助于我们为实际应用选择适当的变体。第三，对标度指数优化的需求揭示了如何加强极化码设计的思路。在本文中，我们将最简通信信道——二元删除信道（BEC），也是众多编码理论研究的核心对象——推广为“四面体删除信道”（TEC）。随后，我们利用Mori-Tanaka在GF$(4)$上的$2 \times 2$矩阵在TEC上构造极化码。我们的主要贡献在于证明TEC的动态收敛到几乎单参数信道族，进而得出标度指数的上界为$3.328$。这是首个获得标度指数上界的非二元矩阵，并且其极化BEC的速度快于所有已知的$23 \times 23$规模的二元矩阵。我们的结果表明，为了实现更快的极化，扩展字母表比扩大矩阵规模更有效且更具实践可行性。