The Hermitian dual codes of several classes of BCH codes

As a special subclass of cyclic codes, BCH codes are usually among the best cyclic codes and have wide applications in communication and storage systems and consumer electronics. Let C be a q 2 -ary BCH code of length n with respect to an n -th primitive root of unity β over an extension field of F q 2 , and let C ⊥ H denote its Hermitian dual code, where q is a prime power. If both C and C ⊥ H are a BCH code with respect to an n -th primitive root of unity β, then C is called a Hermitian dually-BCH code . The objective of this paper is to derive a necessary and sufficient condition for ensuring that two classes of narrow-sense BCH codes are Hermitian dually-BCH codes. As by-products, lower bounds on the minimum distances of the Hermitian dual codes of these BCH codes are developed, which improve the lower bounds documented in IEEE Trans. Inf. Theory, vol. 68, no. 2, pp. 953-964, 2022, in some cases.