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BCH codes are an interesting class of cyclic codes due to their efficientencoding and decoding algorithms. In the past sixty years, a lot of progress onthe study of BCH codes has been made, but little is known about the propertiesof their duals. Recently, in order to study the duals of BCH codes and thelower bounds on their minimum distances, a new concept called dually-BCH codewas proposed by authors in \cite{GDL21}. In this paper, the lower bounds on theminimum 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 apositive integer satisfying $\lambda\, |\, q-1$, or $\lambda=q^s-1$ and $s\,|\,m$. In addition, the sufficient and necessary conditions in terms of thedesigned distances for these codes being dually-BCH codes are presented. Manyconsidered codes in \cite{GDL21} and \cite{Wang23} are the special cases of thecodes showed in this paper.  Our lower bounds on the minimum distances of the duals of BCH codes includethe bounds stated in \cite{GDL21} as a special case. Several examples show thatthe lower bounds are good in some cases.

The duals of narrow-sense BCH codes with length $\frac{q^m-1}{\lambda}$