  Cyclic BCH codes and negacyclic BCH codes form important subclasses of cyclic
codes and negacyclic codes, respectively, and can produce optimal linear codes
in many cases. To the best of our knowledge, there are few results on the dual
codes of cyclic and negacyclic BCH codes. In this paper, we study the dual
codes of narrow-sense cyclic BCH codes of length $q^m+1$ over a finite field
$\mathbb{F}_q$, where $q$ is an odd prime power, and the dual codes of
narrow-sense negacyclic BCH codes of length $\frac{q^{m}+1}{2}$ over
$\mathbb{F}_q$, where $q$ is an odd prime power satisfying $q\equiv 3~({\rm
mod}~4)$. Some lower bounds on the minimum distances of the dual codes are
established, which are very close to the true minimum distances of the dual
codes in many cases. Sufficient and necessary conditions for the even-like
subcodes of narrow-sense cyclic BCH codes of length $q^{m}+1$ being cyclic
dually-BCH codes are given in terms of designed distances, where $q$ is odd and
$m$ is odd or $m\equiv 2~({\rm mod~}4)$. The concept of negacyclic dually-BCH
codes is proposed, and sufficient and necessary conditions in terms of designed
distances are presented to ensure that narrow-sense negacyclic BCH codes of
length $\frac{q^{m}+1}{2}$ are dually-BCH codes, where $q\equiv 3~({\rm
mod}~4)$.
