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 ≥ 4, let n = ^q_q^m₊₁⁻¹ (m even), or n = ^q_q^m₋⁻₁¹ (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 = ^q_q^m₊₁⁻¹ (m even). As byproducts, we present the largest coset leader δ1 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 δ1. The BCH codes presented in this paper have good parameters in general.