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Erschienen in:
01.02.2020
Some negacyclic BCH codes and quantum codes
Erschienen in: Quantum Information Processing | Ausgabe 2/2020
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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.