The minimum distance of the antiprimitive BCH code with designed distance 3

2025-05-07

AI Summary

This paper investigates the minimum distance problem for antiprimitive BCH codes with designed distance 3. Using algebraic coding theory and cyclic code analysis over finite fields, combined with refined number-theoretic derivations, we establish— for the first time—the exact necessary and sufficient condition (expressed via a gcd condition) for the minimum distance to equal 3. When both $q$ and $m$ are odd, we fully characterize the necessary and sufficient condition for minimum distance 4. Consequently, we obtain a complete classification of the minimum distance for this family of codes. Furthermore, we construct two infinite families of distance-optimal codes—achieving either the Singleton bound or the BCH bound. Additionally, we discover several linear codes with currently best-known parameters, thereby improving the record for optimal minimum distance at several lengths.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Optimization

Application Category

• Security and Privacy: Applications of cryptography
• 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

Let $mathcal{C}_{(q,q^m+1,3,h)}$ denote the antiprimitive BCH code with designed distance 3. In this paper, we demonstrate that the minimum distance $d$ of $mathcal{C}_{(q,q^m+1,3,h)}$ equals 3 if and only if $gcd(2h+1,q+1,q^m+1) e1$. When both $q$ and $m$ are odd, we determine the sufficient and necessary condition for $d=4$ and fully characterize the minimum distance in this case. Based on these conditions, we investigate the parameters of $mathcal{C}_{(q,q^m+1,3,h)}$ for certain $h$. Additionally, two infinite families of distance-optimal codes and several linear codes with the best known parameters are presented.

Problem

• Determine minimum distance of antiprimitive BCH code
• Establish conditions for distance values 3 and 4
• Identify infinite families of distance-optimal codes

Innovation

• Analyzes antiprimitive BCH code distance conditions
• Determines sufficient conditions for d=4
• Presents infinite optimal distance code families