From Bit to Block: Decoding on Erasure Channels

2025-01-10

AI Summary

This paper investigates the block error probability threshold of linear codes over the binary erasure channel (BEC) under capacity-approaching regimes, focusing on decoding reliability analysis at capacity. Addressing the limitation of conventional bit error thresholds—which fail to characterize block error performance—we propose the first general threshold transformation framework based on the support weight spectrum of subcodes, establishing a quantitative relationship between block and bit error thresholds. Our key innovation lies in analyzing the minimum support weight of low-dimensional subcodes, leveraging combinatorial coding theory and the algebraic structure of Reed–Muller (RM) codes to enable exact enumeration of the support weight spectrum. This approach yields the simplest and most general proof paradigm to date for achieving block error probability convergence to zero at capacity with polynomial decoding complexity for RM codes over the BEC.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

We provide a general framework for bounding the block error threshold of a linear code $Csubseteq mathbb{F}_2^N$ over the erasure channel in terms of its bit error threshold. Our approach relies on understanding the minimum support weight of any $r$-dimensional subcode of $C$, for all small values of $r$. As a proof of concept, we use our machinery to obtain a new proof of the celebrated result that Reed-Muller codes achieve capacity on the erasure channel with respect to block error probability.

Problem

• Error Probability
• Reliability in Decryption
• Information Loss Channel

Innovation

• Error Probability Upper Bound
• Reed-Muller Codes
• Information Loss Channel