Negacyclic BCH codes of length \(\frac{q^{2m}-1}{q+1}\) and their duals

Negacyclic BCH codes are an important subclass of negacyclic codes and have good parameters. Inspired by the recent work on cyclic codes published in Wu et al. (Finite Fields Appl 60:101581, 2019), the objective of this paper is to investigate the parameters of the narrow-sense negacyclic BCH codes of length \(n=\frac{q^{2m}-1}{q+1}\) $n=\frac{q^{2m}-1}{q+1}$ over \({\textrm{GF}}(q)\) $\text{GF}(q)$ , where q is an odd prime power. For \(2\le \delta \le \left\lfloor \frac{q^{m+1}-q}{q+1} \right\rfloor +2\) $2\le \delta \le ⌊\frac{q^{m+1}-q}{q+1}⌋+2$ , the dimension of the narrow-sense negacyclic BCH codes of length n with designed distance \(\delta \) $\delta$ is determined. For \(2\le \delta \le \frac{q^{2m-1}+1}{q+1}\) $2\le \delta \le \frac{q^{2m-1}+1}{q+1}$ , a lower bound on the minimum distance of the dual codes of the narrow-sense negacyclic BCH codes of length n with designed distance \(\delta \) $\delta$ is presented. Compared with cyclic codes, we obtain some negacyclic BCH codes with better parameters.