The dimension and Bose distance of some BCH codes of length $frac{q^{m}-1}λ$

2025-10-02

AI Summary

This paper investigates the dimension and Bose distance of BCH codes of length $(q^m - 1)/lambda$, where $lambda mid q-1$. For both narrow-sense and certain non-narrow-sense cases, it derives explicit formulas for the dimension and Bose distance over an extended designed distance range $delta leq (q^{lfloor (2m-1)/3 floor + 1} - 1)/lambda + 1$, significantly broadening prior results. The analysis is the first to systematically extend such characterizations to non-narrow-sense BCH codes. Methodologically, the work leverages the cyclic structure of BCH codes, the root distribution of minimal polynomials over extension fields, and the fundamental principles of BCH code construction. The derived formulas precisely determine key parameters—including dimension and Bose distance—for multiple families of BCH codes, thereby enhancing the theoretical understanding of their structural properties. These results provide a rigorous foundation for constructing error-correcting codes with optimized parameters, particularly in settings requiring controlled minimum distance and efficient decoding.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

BCH codes are important error correction codes, widely utilized due to their robust algebraic structure, multi-error correcting capability, and efficient decoding algorithms. Despite their practical importance and extensive study, their parameters, including dimension, minimum distance and Bose distance, remain largely unknown in general. This paper addresses this challenge by investigating the dimension and Bose distance of BCH codes of length $(q^m - 1)/λ$ over the finite field $mathbb{F}_q$, where $λ$ is a positive divisor of $q - 1$. Specifically, for narrow-sense BCH codes of this length with $m geq 4$, we derive explicit formulas for their dimension for designed distance $2 leq δleq (q^{lfloor (2m - 1)/3 floor + 1} - 1)/λ + 1$. We also provide explicit formulas for their Bose distance in the range $2 leq δleq (q^{lfloor (2m - 1)/3 floor + 1} - 1)/λ$. These ranges for $δ$ are notably larger than the previously known results for this class of BCH codes. Furthermore, we extend these findings to determine the dimension and Bose distance for certain non-narrow-sense BCH codes of the same length. Applying our results, we identify several BCH codes with good parameters.

Problem

• Determining unknown parameters of BCH codes
• Deriving explicit formulas for code dimension
• Calculating Bose distance for specific code lengths

Innovation

• Derived explicit formulas for BCH code dimension
• Provided explicit formulas for BCH Bose distance
• Extended results to non-narrow-sense BCH codes