Reed-Muller codes have vanishing bit-error probability below capacity: a simple tighter proof via camellia boosting

TL;DR

This paper introduces a new, simpler proof technique for Reed-Muller codes showing they achieve vanishing bit-error probability below channel capacity, using camellia codes and Fourier analysis.

Contribution

It provides a tighter, more straightforward proof of the bit-error probability results for Reed-Muller codes on symmetric channels, improving upon prior methods.

Findings

Reed-Muller codes have vanishing bit-error probability below capacity.

The proof uses camellia codes and second moment Fourier analysis.

It offers an exponentially tighter bound for error channels.

Abstract

This paper shows that a class of codes such as Reed-Muller (RM) codes have vanishing bit-error probability below capacity on symmetric channels. The proof relies on the notion of `camellia codes': a class of symmetric codes decomposable into `camellias', i.e., set systems that differ from sunflowers by allowing for scattered petal overlaps. The proof then follows from a boosting argument on the camellia petals with second moment Fourier analysis. For erasure channels, this gives a self-contained proof of the bit-error result in Kudekar et al.'17, without relying on sharp thresholds for monotone properties Friedgut-Kalai'96. For error channels, this gives a shortened proof of Reeves-Pfister'23 with an exponentially tighter bound, and a proof variant of the bit-error result in Abbe-Sandon'23. The control of the full (block) error probability still requires Abbe-Sandon'23 for RM codes.