The Duals of Narrow-Sense BCH Codes With Length qm-1/λ

BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In the past sixty years, a lot of progress on the study of BCH codes has been made, but little is known about the properties of their duals. Recently, in order to study the duals of BCH codes and the lower bounds on their minimum distances, a new concept called dually-BCH code was proposed by (Gong et al., 2022). In this paper, the lower bounds on the minimum distances of the m - 1 duals of narrow-sense BCH codes with length q lambda over Fq are developed, where lambda is a positive integer satisfying lambda = q s - 1 and s m , or lambda q - 1. In addition, the sufficient and necessary conditions in terms of the designed distances for these codes being dually-BCH codes are presented. Our lower bounds on the minimum distances of the duals of BCH codes include the bounds stated in (Gong et al., 2022) as a special case. Moreover, our lower bounds improve the bounds stated in (Gong et al., 2022), the classical Sidel'nikov bound, and the Carlitz-Uchiyama bound when the designed distances of the BCH codes are in some ranges. Several examples show that our proposed lower bounds are good in some cases.