On the Worst-Case Complexity of Gibbs Decoding for Reed--Muller Codes

TL;DR

This paper investigates the Gibbs decoder for Reed--Muller codes, revealing that it can have super-polynomial worst-case mixing times, which impacts its efficiency in capacity-achieving decoding.

Contribution

It provides the first analysis of the Gibbs decoder's mixing time for Reed--Muller codes, showing potential super-polynomial complexity in worst-case scenarios.

Findings

Gibbs decoder can exhibit slow mixing for certain sequences.

Slow mixing implies super-polynomial decoding time in worst cases.

Analysis highlights limitations of MCMC-based decoding methods.

Abstract

Reed--Muller (RM) codes are known to achieve capacity on binary symmetric channels (BSC) under the Maximum a Posteriori (MAP) decoder. However, it remains an open problem to design a capacity achieving polynomial-time RM decoder. Due to a lemma by Liu, Cuff, and Verd\'u, it can be shown that decoding by sampling from the posterior distribution is also capacity-achieving for RM codes over BSC. The Gibbs decoder is one such Markov Chain Monte Carlo (MCMC) based method, which samples from the posterior distribution by flipping message bits according to the posterior, and can be modified to give other MCMC decoding methods. In this paper, we analyze the mixing time of the Gibbs decoder for RM codes. Our analysis reveals that the Gibbs decoder can exhibit slow mixing for certain carefully constructed sequences. This slow mixing implies that, in the worst-case scenario, the decoder requires super-polynomial time to converge to the desired posterior distribution.