BCH codes and their dual codes are two special subclasses of cyclic codes and are the best linear codes in many cases. A lot of progress on the study of BCH cyclic codes has been made, but little is known about the minimum distances of the duals of BCH codes. Recently, a new concept called dually-BCH code was introduced to investigate the duals of BCH codes and the lower bounds on their minimum distances in \cite{GDL21}. For a prime power $q$ and an integer $m \ge 4$, let $n=\frac{q^m-1}{q+1}$ \ ($m$ even), or $n=\frac{q^m-1}{q-1}$ \ ($q>2$).
In this paper, some sufficient and necessary conditions in terms of the designed distance will be given to ensure that the narrow-sense BCH codes of length $n$ are dually-BCH codes, which extended the results in \cite{GDL21}. Lower bounds on the minimum distances of their dual codes are developed for $n=\frac{q^m-1}{q+1}$ \ ($m$ even). As byproducts, we present the largest coset leader $δ_1$ modulo $n$ being of two types, which proves a conjecture in \cite{WLP19} and partially solves an open problem in \cite{Li2017}. We also investigate the parameters of the narrow-sense BCH codes of length $n$ with design distance $δ_1$. The BCH codes presented in this paper have good parameters in general.
A Structural Characterization of Cyclotomic Cosets with Applications to Affine-Invariant Codes and BCH Codes
Hubei University（湖北大学）；Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University（湖北大学数学与统计学院应用数学湖北省重点实验室）；Key Laboratory of Intelligent Sensing System and Security (Hubei University), Ministry of Education（教育部智能感知系统与安全性重点实验室（湖北大学））
Hermitian dual-containing constacyclic BCH codes and related quantum codes of length $\frac{q^{2m}-1}{q+1}$
本文研究了$\mathbb{F}_{q^2}$上的一类长度为$n=\frac{q^{2m}-1}{q+1}$的常循环BCH码，其中$q$是素数幂，$m\geq2$是偶数.确定了长度为$n$的窄意义厄米特对偶含常循环BCH码的最大设计距离。进一步计算了给定设计距离的常循环BCH码的精确维数。因此，我们能够推导出量子码的参数作为相关的常循环BCH码的设计参数的函数。这改进了Yuan等人的结果（Des Codes Cryptogr 85（1）：179-190，2017），表明在相同长度的情况下，除了三种平凡情况（$q=2,3,4$），我们得到的量子码总是可以产生比Yuan等人获得的量子码严格的维度或最小距离增益。此外，固定长度$n=\frac{q^{2m}-1}{q+1}$，与一些已知结果相比，一些构造的量子码具有更好的参数或者是有益的补充（Aly等人，IEEE Trans Inf Theory 53（3）：1183-1188，2007，Li等人，Quantum Inf Process 18（5）：127，2019，Wang等人，Quantum Inf Process 18（8）：323，2019，Song等人，Quantum Inf Process 17（10）：1-24，2018.）。
On the Minimum Distances of Some Families of BCH Codes
College of Cyber Security, Jinan University（济南大学网络空间安全学院）；College of Information Science and Technology, Jinan University（济南大学信息科学与技术学院）；Department of Computer Science and Engineering, The Hong Kong University of Science and Technology（香港科技大学计算机科学与工程系）；Strategic Centre for Research in Privacy-Preserving Technologies and Systems, Nanyang Technological University（南洋理工大学隐私保护技术与系统研究中心）
