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

TL;DR

This paper demonstrates that Reed--Muller codes, with some random scrambling, can reliably achieve the symmetric capacity of finite-state channels by leveraging symmetry properties and code interleaving techniques.

Contribution

It extends symmetry-based capacity results from memoryless channels to finite-state channels using Reed--Muller codes and a novel reduction approach.

Findings

Reed--Muller codes achieve symmetric capacity on finite-state channels.

A capacity-via-symmetry theorem for group codes on DMCs is established.

Binary Reed--Muller codes can be used to construct non-binary codes for FSCs.

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.