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

2025-05-02

AI Summary

This work investigates the worst-case computational complexity of Gibbs decoding for Reed–Muller (RM) codes over the binary symmetric channel (BSC), focusing on the mixing time of the underlying Markov chain. Leveraging tools from Markov chain Monte Carlo (MCMC) theory, posterior distribution sampling analysis, and combinatorial code construction, we establish the first rigorous proof that there exist specific sequences of RM codewords for which the Gibbs decoder’s mixing time grows superpolynomially in block length. This result exposes a fundamental limitation: Gibbs decoding cannot guarantee capacity-achieving performance in polynomial time for all RM codes, thereby refuting the common assumption of universal efficient convergence. The core contribution is the derivation of the first worst-case lower bound on mixing time for Gibbs decoding of RM codes—providing a critical theoretical criterion for the computational feasibility of sampling-based decoders for structured code families.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Analyzing Gibbs decoder mixing time for RM codes
• Identifying slow mixing in worst-case RM code sequences
• Establishing super-polynomial convergence time for Gibbs decoder

Innovation

• Gibbs decoder samples posterior distribution
• MCMC method flips message bits
• Analyzes mixing time for RM codes