Two classes of LCD BCH codes over finite fields

BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths $\frac{q^m+1}{q+1}$ and $q^m+1$ over the finite field $F_q$, both of which are LCD codes. The dimensions of narrow-sense BCH codes of length $\frac{q^m+1}{q+1}$ with designed distance $\delta =\ell q^{\frac{m-1}{2}}+1$ are determined, where $q>2$ and $2\le \ell \le q-1$. Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length $q^m+1$ are developed for odd q, which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length $q^m+1$ being dually-BCH codes are presented, where q is odd and $m\not\equiv 0\left(\mathrm{mod}4\right)$.