Reed-Muller Codes Achieve Capacity on Erasure Channels

TL;DR

This paper proves that Reed-Muller codes and similar affine-invariant codes achieve channel capacity on erasure channels by leveraging code symmetry, extending previous results to a broader class of codes.

Contribution

It introduces a symmetry-based method to prove capacity achievement for a wide class of linear codes, including Reed-Muller and affine-invariant codes, under maximum a posteriori decoding.

Findings

Reed-Muller codes achieve capacity on erasure channels.

Symmetry alone implies near-optimal performance for certain codes.

Extends capacity-achieving results to affine-invariant and primitive narrow-sense BCH codes.

Abstract

We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes our method exploits code symmetry. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive. In other words, we show that symmetry alone implies near-optimal performance. An important consequence of this result is that a sequence of Reed-Muller codes with increasing blocklength and converging rate achieves capacity. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to all affine-invariant codes and, thus, to extended primitive narrow-sense BCH codes. This also resolves, in the affirmative, the existence question for capacity-achieving sequences of binary cyclic codes. The primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions.