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

2026-04-16

AI Summary

This work investigates reliable communication achieving the symmetric capacity over indecomposable finite-state channels (FSCs) with binary inputs. To this end, the authors propose a coding scheme based on a single binary Reed–Muller (RM) code, which—through random scrambling, bit grouping, and nonbinary interleaving—effectively transforms the original channel with memory into an approximately memoryless nonbinary channel. By leveraging the symmetry properties of doubly transitive group codes and invoking the symmetric capacity theorem, they rigorously prove that the proposed scheme can approach and attain the symmetric capacity of the FSC. This study establishes, for the first time, the capacity-achieving capability of RM codes over finite-state channels, thereby significantly extending their theoretical applicability to channels with memory.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Information Theory
• Multiagent Systems: Agent Communication

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Reed-Muller codes
• finite-state channels
• symmetric capacity
• reliable communication
• uniform-input information rate

Innovation

• Reed-Muller codes
• finite-state channels
• symmetric capacity
• doubly-transitive group codes
• interleaving