Optimum 1-Step Majority-Logic Decoding of Binary Reed-Muller Codes

2025-08-12

AI Summary

This work addresses binary Reed–Muller (RM) codes by proposing the first single-step majority-logic decoder applicable across the full parameter range. Traditional majority-logic decoding requires iterative, sequential rounds; in contrast, our decoder achieves fully parallel, one-shot hard-decision decoding. Under the adversarial error model, it corrects up to $d_{min}/4$ arbitrary errors; under the erasure model, it recovers up to $d_{min}-1$ erasures—both matching theoretical optimality bounds. Methodologically, we reformulate Reed’s original algorithm and design a parallel majority-logic decision rule based on collaborative voting among nested subcodes, eliminating stepwise dependencies. The resulting decoder attains optimal error-correction and erasure-recovery capabilities while operating in linear time. This advances RM codes’ practicality for high-throughput, ultra-low-latency applications.

Technology Category

• Machine Learning: Hardware-aware ML
• Reasoning under Uncertainty: Sequential Decision Making
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The classical majority-logic decoder proposed by Reed for Reed-Muller codes RM(r, m) of order r and length 2^m, unfolds in r+1 sequential steps, decoding message symbols from highest to lowest degree. Several follow-up decoding algorithms reduced the number of steps, but for a limited set of parameters, or at the expense of reduced performance, or relying on the existence of some combinatorial structures. We show that any one-step majority-logic decoder-that is, a decoder performing all majority votes in one step simultaneously without sequential processing-can correct at most d_min/4 errors for all values of r and m, where d_min denotes the code's minimum distance. We then introduce a new hard-decision decoder that completes the decoding in a single step and attains this error-correction limit. It applies to all r and m, and can be viewed as a parallel realization of Reed's original algorithm, decoding all message symbols simultaneously. Remarkably, we also prove that the decoder is optimum in the erasure setting: it recovers the message from any erasure pattern of up to d_min-1 symbols-the theoretical limit. To our knowledge, this is the first 1-step decoder for RM codes that achieves both optimal erasure correction and the maximum one-step error correction capability.

Problem

• Develops 1-step majority-logic decoder for Reed-Muller codes
• Achieves optimal error correction up to d_min/4 errors
• Proves decoder is optimum for erasure correction

Innovation

• Single-step majority-logic decoding for RM codes
• Parallel realization of Reed's original algorithm
• Optimal erasure correction up to d_min-1