For the polar codes introduced by Arikan in 2009, the first code family achieving the capacity of binary-input discrete memoryless channels (BIDMCs) with low-complexity encoding and decoding, it is crucial to evaluate the reliability of the synthetic channels resulted in the code construction. Since the synthetic channels have an output alphabet that grows exponentially with the code length, an effective method for faithfully evaluating their reliability is to replace them with degradations of manageable alphabet size. The main aim of this paper is to find the optimal degradations of symmetric BIDMCs. We determine all the degradations with the minimum probability of error decoding and give a necessary condition for the degradations with the maximum symmetric capacity. Finally, based on this necessary condition, we propose an efficient algorithm for finding degradation schemes that maximize the symmetric capacity.

翻译：针对Arikan于2009年提出的极化码——首个具备低复杂度编解码能力、可实现二元输入离散无记忆信道(BIDMC)容量的编码族，评估编码构造中合成信道的可靠性至关重要。由于合成信道的输出字母表随码长呈指数级增长，对其进行可靠评估的有效方法是用具有可控字母表大小的退化信道替代合成信道。本文主要旨在寻找对称BIDMC的最优退化方案。我们确定了所有具有最小译码错误概率的退化方案，并给出了具有最大对称容量的退化方案的必要条件。最后，基于该必要条件，我们提出了一种能够最大化对称容量的退化方案高效求解算法。