  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 $\delta_1$ 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 $\delta_1$. The BCH codes presented in this paper have good
parameters in general.
