Some negacyclic BCH codes and quantum codes

Abstract

In this paper, we investigate narrow-sense and non-narrow-sense negacyclic Bose–Chaudhuri–Hocquenghem (NBCH) codes of length $n=\frac{q^m-1}{a}(q^m+1)$ over ${\mathbb {F}}_{q^2}$ closely, where q is an odd prime power, $m\ge 3$ is an odd integer and $a\mid (q^m-1)$ is an even integer. To derive accurate maximum designed distance of Hermitian dual containing NBCH codes, we define $2\le a\le 2q^2-q-1$ for narrow-sense codes with $\delta _{m, a}^N$ and $2\le a< 2(q-1)$ for non-narrow-sense codes with $\delta _{m, a}^{NN}$. For given a, our maximum designed distance improves over the distance $\delta _m^A$ of Aly et al. (IEEE Trans Inf Theory 53:1183–1188, 2007) to a great extent, that is, $\delta _{m, a}^{N}=\delta _{m, a}^{NN}=\frac{a+2}{2}\delta _m^A$. After determining dimensions of such Hermitian dual containing NBCH codes, we construct many new quantum codes via Hermitian construction naturally, whose parameters are better than the ones in the literature.