On the burst-covering radius of binary cyclic codes

2026-01-01

arXiv.org

AI Summary

This study investigates the burst error covering capability of binary cyclic codes and introduces the novel concept of “burst covering radius” to characterize the extent to which codewords cover all possible burst error patterns. By analyzing pattern frequencies in linear feedback shift register (LFSR) sequences, the authors establish a connection between these sequences and the algebraic structure of cyclic codes, deriving a general upper bound on the burst covering radius. Notably, tighter bounds are obtained for the first time for primitive BCH codes and Melas codes. Building on these theoretical results, the paper further presents an efficient algorithm for constructing cyclic codes with enhanced burst covering performance. Integrating algebraic coding theory, LFSR sequence analysis, and combinatorial bounding techniques, this work provides new theoretical tools and practical solutions for code design in burst error-prone environments.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We define and study burst-covering codes. We provide some general bounds connecting the code parameters with its burst-covering radius. We then provide stronger bounds on the burst-covering radius of cyclic codes, by employing linear-feedback shift-register (LFSR) sequences. For the case of BCH codes we prove a new bound on pattern frequencies in LFSR sequences, which is of independent interest. Using this tool, we can bound the covering-radius of binary primitive BCH codes and Melas codes. We conclude with an efficient algorithm for burst-covering cyclic codes.

Problem

• burst-covering radius
• binary cyclic codes
• BCH codes
• covering radius
• LFSR sequences

Innovation

• burst-covering radius
• cyclic codes
• LFSR sequences
• BCH codes
• covering radius