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arXiv · 2604.15295

Reed--Muller Codes Achieve the Symmetric Capacity on Finite-State Channels

Henry D. Pfister·Navin Kashyap·Jean-Francois Chamberland·Galen Reeves
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Abstract

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.

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Henry D. Pfister, Navin Kashyap, Jean-Francois Chamberland, Galen Reeves. 2026-04-16. Reed--Muller Codes Achieve the Symmetric Capacity on Finite-State Channels. https://arxiv.org/abs/2604.15295

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