  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.
