BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In the past sixty years, a lot of progress on the study of BCH codes has been made, but little is known about the properties of their duals. Recently, in order to study the duals of BCH codes and the lower bounds on their minimum distances, a new concept called dually-BCH code was proposed by authors in \cite{GDL21}. In this paper, the lower bounds on the minimum distances of the duals of narrow-sense BCH codes with length $\frac{q^m-1}{\lambda}$ over $\mathbb{F}_q$ are developed, where $\lambda$ is a positive integer satisfying $\lambda\, |\, q-1$, or $\lambda=q^s-1$ and $s\, |\,m$. In addition, the sufficient and necessary conditions in terms of the designed distances for these codes being dually-BCH codes are presented. Many considered codes in \cite{GDL21} and \cite{Wang23} are the special cases of the codes showed in this paper. Our lower bounds on the minimum distances of the duals of BCH codes include the bounds stated in \cite{GDL21} as a special case. Several examples show that the lower bounds are good in some cases.

翻译：BCH码因具备高效的编码与译码算法而成为一类引人关注的循环码。过去六十年间，尽管BCH码的研究取得了诸多进展，但其对偶码的性质仍鲜为人知。近期为探究BCH码对偶码及其最小距离下界，文献\cite{GDL21}作者提出了"对偶BCH码"这一新概念。本文建立了有限域$\mathbb{F}_q$上长度为$\frac{q^m-1}{\lambda}$的狭义BCH码对偶码的最小距离下界，其中$\lambda$为满足$\lambda\, |\, q-1$或$\lambda=q^s-1$且$s\, |\,m$的正整数。此外，本文给出了这类码成为对偶BCH码时关于设计距离的充要条件。文献\cite{GDL21}与\cite{Wang23}中讨论的诸多码型均为本文所述码型的特例。本文给出的BCH码对偶码最小距离下界将文献\cite{GDL21}的结论作为特例包含其中。多个实例表明，这些下界在某些情况下具有良好性能。