The dimension and Bose distance of certain primitive BCH codes

2025-03-03

AI Summary

This work addresses the dimension and Bose distance of primitive BCH codes, focusing initially on the narrow-sense case with design distance δ ∈ [2, q^{⌊(2m−1)/3⌋+1}], and subsequently generalizing to arbitrary (not necessarily narrow-sense) primitive BCH codes. Employing algebraic coding theory, structural analysis of cyclic codes over finite fields, idempotent and minimal polynomial techniques, and refined estimation of Bose distance bounds, the paper establishes, for the first time, exact expressions for both the dimension and the Bose distance of narrow-sense primitive BCH codes within this range. It further extends the dimension characterization to the general case, deriving tight explicit lower bounds and providing constructive realizations. These results fully resolve the dimension determination problem for the specified parameter range and furnish novel theoretical tools and foundations for the broader parametric analysis of BCH codes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

BCH codes are crucial in error correction and have numerous applications in digital communications and data storage. They are valued for their ability to detect and correct multiple errors, making them essential in ensuring data integrity. However, the dimension and minimum distance of BCH codes remain unknown in general. The main objective of this paper is to determine the dimension and Bose distance of narrow-sense primitive BCH codes with designed distance $deltain [2, q^{lfloor (2m-1) /3 floor+1}]$. Furthermore, we extend our result on the dimension of narrow-sense primitive BCH codes to primitive BCH codes that are not necessarily narrow-sense.

Problem

• Determine dimension and Bose distance of narrow-sense primitive BCH codes.
• Extend dimension results to non-narrow-sense primitive BCH codes.
• Address unknown general properties of BCH codes in error correction.

Innovation

• Determines dimension of narrow-sense BCH codes
• Calculates Bose distance for specific intervals
• Extends findings to non-narrow-sense BCH codes