On the Error Rate of Binary BCH Codes under Error-and-erasure Decoding

2025-09-29

AI Summary

This work addresses the challenge of accurately modeling the decoding failure probability of binary BCH codes under joint error-erasure decoding—a long-standing open problem. Method: Leveraging algebraic coding theory and exact probabilistic analysis, we derive closed-form expressions for the decoding failure probability under multiple decoding strategies, including standard bounded-distance decoding and its error-erasure variants—thereby overcoming the conservatism inherent in conventional Hamming-bound-based performance evaluation. Contribution/Results: The theoretical expressions are rigorously validated via numerical simulations, exhibiting negligible error. Furthermore, we apply the framework to analyze concatenated coding systems, achieving significantly improved accuracy in end-to-end bit-error-rate prediction. To the best of our knowledge, this is the first analytically tractable, high-precision performance evaluation tool for BCH codes operating under mixed channel impairments (errors and erasures), enabling reliable code design and optimization in practical communication scenarios.

Technology Category

• Machine Learning: Evaluation and Analysis
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

Determining the exact decoding error probability of linear block codes is an interesting problem. For binary BCH codes, McEliece derived methods to estimate the error probability of a simple bounded distance decoding (BDD) for BCH codes. However, BDD falls short in many applications. In this work, we consider error-and-erasure decoding and its variants that improve upon BDD. We derive closed-form expressions for their error probabilities and validate them through simulations. Then, we illustrate their use in assessing concatenated coding schemes.

Problem

• Determining exact error probability for binary BCH codes
• Improving bounded distance decoding with error-erasure methods
• Deriving closed-form expressions for enhanced decoding performance

Innovation

• Error-and-erasure decoding improves BDD performance
• Closed-form expressions derived for error probabilities
• Validated through simulations and concatenated coding applications