Two classes of narrow-sense BCH codes and their duals
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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 <cit.>. For a prime power q and an integer m ≥ 4, let n=q^m-1/q+1 (m even), or n=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 <cit.>. Lower bounds on the minimum distances of their dual codes are developed for n=q^m-1/q+1 (m even). As byproducts, we present the largest coset leader δ_1 modulo n being of two types, which proves a conjecture in <cit.> and partially solves an open problem in <cit.>. We also investigate the parameters of the narrow-sense BCH codes of length n with design distance δ_1. The BCH codes presented in this paper have good parameters in general.
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