Two Classes of Narrow-Sense BCH Codes and Their Duals
Xiaoqiang Wang, Jiaojiao Wang, Chengju Li, Yansheng Wu · IEEE Transactions on Information Theory · 2023
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.