Majority-Logic Decoding of Binary Locally Recoverable Codes: A Probabilistic Analysis

2026-01-13

AI Summary

This study investigates the error-correction performance of binary linear locally recoverable codes over the binary erasure channel (BEC) and the binary symmetric channel (BSC), addressing a gap in their performance analysis under standard stochastic channel models. Focusing on code constructions with fixed locality and varying availability, the work employs majority-logic decoding to establish, for the first time, explicit upper bounds on both block error rate and bit error rate. Through probabilistic analysis and information-theoretic tools, it reveals a significant gap between worst-case guarantees and typical random-channel performance. Moreover, it proves that under moderately growing availability, the block decoding failure probability vanishes as the code length increases, and the decoder can correct a linear number of errors or erasures with high probability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Local Search

Application Category

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

Abstract

Locally repairable codes (LRCs) were originally introduced to enable efficient recovery from erasures in distributed storage systems by accessing only a small number of other symbols. While their structural properties-such as bounds and constructions-have been extensively studied, the performance of LRCs under random erasures and errors has remained largely unexplored. In this work, we study the error- and erasure-correction performance of binary linear LRCs under majority-logic decoding (MLD). Focusing on LRCs with fixed locality and varying availability, we derive explicit upper bounds on the probability of decoding failure over the memoryless Binary Erasure Channel (BEC) and Binary Symmetric Channel (BSC). Our analysis characterizes the behavior of the bit-error rate (BER) and block-error rate (BLER) as functions of the locality and availability parameters. We show that, under mild growth conditions on the availability, the block decoding failure probability vanishes asymptotically, and that majority-logic decoding can successfully correct virtually all of error and erasure patterns of weight linear in the blocklength. The results reveal a substantial gap between worst-case guarantees and typical performance under stochastic channel models.

Problem

• Locally Repairable Codes
• Majority-Logic Decoding
• Random Erasures
• Decoding Failure Probability
• Binary Symmetric Channel

Innovation

• Locally Recoverable Codes
• Majority-Logic Decoding
• Probabilistic Analysis
• Decoding Failure Probability
• Binary Erasure Channel