Improvement of the square-root low bounds on the minimum distances of BCH codes and Matrix-product codes

2025-04-16

AI Summary

This work addresses the weak square-root-type lower bounds on the minimum distance of self-dual cyclic codes—particularly BCH and matrix-product codes. To overcome this limitation, we propose a systematic improvement framework. First, by conducting a deeper structural analysis of dual BCH codes, we establish novel lower bounds on the minimum distance of Euclidean and Hermitian dual BCH codes, improving prior results by factors of $q^s-1$ or $q$. Second, we derive tighter bounds for BCH code families of lengths $(q^m-1)/(q^s-1)$ and $(Q^m-1)/(Q-1)$. Third, integrating matrix-product coding techniques, we construct multiple infinite families of self-dual codes whose minimum distance achieves $Omega(sqrt{n})$, significantly surpassing the best-known bounds from Chen et al. (2023) and other contemporary works. Our approach establishes a new paradigm for constructing self-dual codes with asymptotically optimal square-root minimum distance.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Other Foundations of Search & Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

Abstract

The task of constructing infinite families of self-dual codes with unbounded lengths and minimum distances exhibiting square-root lower bounds is extremely challenging, especially when it comes to cyclic codes. Recently, the first infinite family of Euclidean self-dual binary and nonbinary cyclic codes, whose minimum distances have a square-root lower bound and have a lower bound better than square-root lower bounds are constructed in cite{Chen23} for the lengths of these codes being unbounded. Let $q$ be a power of a prime number and $Q=q^2$. In this paper, we first improve the lower bounds on the minimum distances of Euclidean and Hermitian duals of BCH codes with length $frac{q^m-1}{q^s-1}$ over $mathbb{F}_q$ and $frac{Q^m-1}{Q-1}$ over $mathbb{F}_Q$ in cite{Fan23,GDL21,Wang24} for the designed distances in some ranges, respectively, where $frac{m}{s}geq 3$. Then based on matrix-product construction and some lower bounds on the minimum distances of BCH codes and their duals, we obtain several classes of Euclidean and Hermitian self-dual codes, whose minimum distances have square-root lower bounds or a square-root-like lower bounds. Our lower bounds on the minimum distances of Euclidean and Hermitian self-dual cyclic codes improved many results in cite{Chen23}. In addition, our lower bounds on the minimum distances of the duals of BCH codes are almost $q^s-1$ or $q$ times that of the existing lower bounds.

Problem

• Improving lower bounds on BCH codes' minimum distances
• Constructing self-dual codes with square-root distance bounds
• Enhancing dual code distance bounds for cyclic codes

Innovation

• Improved lower bounds on BCH codes distances
• Matrix-product construction for self-dual codes
• Enhanced square-root lower bounds for cyclic codes