Reed-Muller Codes on BMS Channels Achieve Vanishing Bit-Error Probability for All Rates Below Capacity

TL;DR

This paper proves that Reed-Muller codes can achieve vanishing bit-error probability on BMS channels for all rates below capacity, advancing understanding of their optimality in information theory.

Contribution

It demonstrates that Reed-Muller codes achieve capacity on BMS channels under bit-MAP decoding, resolving a long-standing open problem with a novel proof technique.

Findings

Reed-Muller codes achieve vanishing error probability at rates approaching capacity.

The proof does not rely on hypercontractivity, unlike previous results for erasure channels.

New information inequalities relate extrinsic information transfer and MMSE in the analysis.

Abstract

This paper considers the performance of Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori (bit-MAP) decoding. Its main result is that, for a fixed BMS channel, the family of binary RM codes can achieve a vanishing bit-error probability at rates approaching the channel capacity. This partially resolves a long-standing open problem that connects information theory and error-correcting codes. In contrast with the earlier result for the binary erasure channel, the new proof does not rely on hypercontractivity. Instead, it combines a nesting property of RM codes with new information inequalities relating the generalized extrinsic information transfer function and the extrinsic minimum mean-squared error.