Improvement of the square-root low bounds on the minimum distances of BCH codes and Matrix-product codes
Xiaoqiang Wang, Liuyi Li, Yansheng Wu, Dabin Zheng, Shuxian Lu
TL;DR
The paper addresses the challenge of constructing infinite families of self-dual codes with unbounded length and square-root-type minimum distances, focusing on BCH codes. It refines dual-distance bounds for BCH codes over $\mathbb{F}_q$ and Hermitian duals over $\mathbb{F}_Q$, using coset-leader and $q$-adic expansions, and combines these with matrix-product code constructions to produce Euclidean and Hermitian self-dual codes with square-root or square-root-like bounds. The authors obtain new lower bounds on $d$ for $\mathcal{C}_{\delta}^{\perp}$ and $\mathcal{C}_{\delta}^{\perp H}$ that scale with $q^s-1$ or $Q-1$ relative to prior results, and present several infinite families of self-dual codes with improved minimum distances, often outperforming Chen23. These results advance the design of high-distance self-dual cyclic codes and broaden the catalog of practical constructions for large-scale quantum- and classical-error-correcting codes.
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.