The Hermitian dual containing non-primitive BCH codes

Let $q$ be a prime power and $m\ge 4$ an even integer. Suppose that $n = \frac {q^{2m}-1}a$ such that $m$ is the multiplicative order of $q^{2}$ modulo $n$ , where $a\ge 2$ is a positive integer. This letter mainly determines the maximum designed distance of Hermitian dual-containing Bose-Chaudhuri-Hocquenghem (BCH) codes of length $n$ over $\Bbb F_{q^{2}}$ . Our results show that the designed distances of non-primitive BCH codes in this letter are larger. Moreover, we obtain the dimensions of some non-primitive BCH codes.