In this paper, we investigate narrow-sense and non-narrow-sense negacyclic Bose–Chaudhuri–Hocquenghem (NBCH) codes of length n=qm-1a(qm+1)\documentclass[12pt]{minimal}
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\begin{document}$$n=\frac{q^m-1}{a}(q^m+1)$$\end{document} over Fq2\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {F}}_{q^2}$$\end{document} closely, where q is an odd prime power, m≥3\documentclass[12pt]{minimal}
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\begin{document}$$m\ge 3$$\end{document} is an odd integer and a∣(qm-1)\documentclass[12pt]{minimal}
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\begin{document}$$a\mid (q^m-1)$$\end{document} is an even integer. To derive accurate maximum designed distance of Hermitian dual containing NBCH codes, we define 2≤a≤2q2-q-1\documentclass[12pt]{minimal}
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\begin{document}$$2\le a\le 2q^2-q-1$$\end{document} for narrow-sense codes with δm,aN\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{m, a}^N$$\end{document} and 2≤a<2(q-1)\documentclass[12pt]{minimal}
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\begin{document}$$2\le a< 2(q-1)$$\end{document} for non-narrow-sense codes with δm,aNN\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{m, a}^{NN}$$\end{document}. For given a, our maximum designed distance improves over the distance δmA\documentclass[12pt]{minimal}
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\begin{document}$$\delta _m^A$$\end{document} of Aly et al. (IEEE Trans Inf Theory 53:1183–1188, 2007) to a great extent, that is, δm,aN=δm,aNN=a+22δmA\documentclass[12pt]{minimal}
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\begin{document}$$\delta _{m, a}^{N}=\delta _{m, a}^{NN}=\frac{a+2}{2}\delta _m^A$$\end{document}. 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.
