We study reliable communication over finite-state channels (FSCs) using Reed--Muller (RM) codes. Building on recent symmetry-based analyses for memoryless channels, we show that a sequence of binary RM codes (with some random scrambling) can achieve the symmetric capacity (or uniform-input information rate) of a binary-input indecomposable FSC. Our approach has three components. First, we establish a capacity-via-symmetry theorem for doubly-transitive group codes on discrete memoryless channels (DMCs) with non-binary inputs, under some symmetry and puncturing conditions. Then, we reduce a binary-input FSC to an almost memoryless non-binary channel by grouping adjacent input bits into blocks and interleaving non-binary codes onto the channel. Finally, we show that the interleaved non-binary codes can be constructed from a single binary RM code.

翻译：本文研究在有限状态信道（FSC）上使用Reed--Muller（RM）码的可靠通信。基于近期对无记忆信道的对称性分析，我们证明，一系列二元RM码（经随机加扰）可在二元输入不可分解FSC上达到对称容量（即均匀输入信息速率）。我们的方法包含三个部分。首先，针对具有非二元输入的离散无记忆信道（DMC），在满足对称性和打孔条件的设定下，建立了双重传递群码的容量对称性定理。接着，通过将相邻输入比特分组为块并在信道上对非二元码进行交织，将二元输入FSC简化为近乎无记忆的非二元信道。最后，证明此交织后的非二元码可由单个二元RM码构造。