On the basis of concepts of skew symmetric -cyclotomic coset and skew asymmetric coset pair, and their properties deeply studied, and methods determined, a maximal design distance of BCH codes with length contained in its Hermitian dual is studied. Thus, such an unsolved problem for maximal design distance of BCH codes with length is solved, upper bound of maximal design distance of BCH codes previously known is improved, and a new upper bound picked up is sharp. By utilizing these Hermitian dual containing non-narrow-sense BCH codes, many new quantum codes with good parameters are constructed, these new quantum codes are better than that constructed from narrow-sense BCH codes in the literature.