Two Classes of Narrow-Sense BCH Codes and Their Duals

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 duals of BCH codes. Recently, a concept called dually-BCH code was introduced to investigate the duals of BCH codes and the lower bounds on their minimum distances in Gong et al., (2022). 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 Gong et al., (2022). 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 Wu et al., (2019) and partially solves an open problem in Li et al., (2017). We also investigate the parameters of narrow-sense BCH codes of length
$n$
with design distance
$\delta _{1}$
. The BCH codes presented in this paper have good parameters in general.