BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers.
The dimension of narrow-sense BCH codes of length $\frac{q^m-1}λ$ with designed distance $2\leq δ\leq \frac{q^{\lceil(m+1)/2 \rceil}-1}λ+1$ is settled, where $λ$ is any factor of $q-1$.
The weight distributions of two classes of narrow-sense BCH codes of length $\frac{q^m-1}2$ with designed distance $δ=\frac{(q-1)q^{m-1}-q^{\lfloor(m-1)/2\rfloor}-1}2$ and $δ=\frac{(q-1)q^{m-1}-q^{\lfloor(m+1)/2\rfloor}-1}2$ are determined.
The weight distribution of a class of BCH codes of length $\frac{q^m-1}{q-1}$ is determined.
In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized.
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）
Existence and Constructions of Strict Function-Correcting Codes with Data Protection
The International Institute of Information Technology Hyderabad（印度国际信息技术学院海得拉巴分校）；Indian Institute of Science, Bengaluru（印度科学研究所班加罗尔）；Aalto University（阿尔托大学）
