Correcting mistakes in the paper “A mass formula for negacyclic codes of length 2k and some good negacyclic codes over $\mathbb {Z}_{4}+u\mathbb {Z}_{4}$” [Cryptogr. Commun. (2017) 9: 241–272]

Chen, Yuan; Cao, Yonglin; Bandi, Ramakrishna; Fu, Fang-Wei

doi:10.1007/s12095-020-00429-z

Cited by 1 publication

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“…In 2018, we established successfully all negacyclic codes of length 4p s over F p m + uF p m [32]. After that, some authors extended these problems to many more general lengths and alphabets (see, e.g., [11], [12], [13], [14], [15], [16], [17], [18], [19], [39], [50]).…”

Section: Introductionmentioning

confidence: 99%

Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $

Dinh

Nguyen

Maneejuk

2022

AMC

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<p style='text-indent:20px;'>For any odd prime <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>, the structures and duals of <inline-formula><tex-math id="M4">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-constacyclic codes of length <inline-formula><tex-math id="M5">\begin{document}$ 8p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M6">\begin{document}$ \mathcal R = \mathbb F_{p^m}+u\mathbb F_{p^m} $\end{document}</tex-math></inline-formula> are completely determined for all unit <inline-formula><tex-math id="M7">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> of the form <inline-formula><tex-math id="M8">\begin{document}$ \lambda = \xi^l\in \mathbb F_{p^m} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M9">\begin{document}$ l $\end{document}</tex-math></inline-formula> is even. In addition, the algebraic structures of all cyclic and negacyclic codes of length <inline-formula><tex-math id="M10">\begin{document}$ 8p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M11">\begin{document}$ \mathcal R $\end{document}</tex-math></inline-formula> are established in term of their generator polynomials. Dual codes of all cyclic and negacyclic codes of length <inline-formula><tex-math id="M12">\begin{document}$ 8p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M13">\begin{document}$ \mathcal R $\end{document}</tex-math></inline-formula> are also investigated. Furthermore, we give the number of codewords in each of those cyclic and negacyclic codes. We also obtain the number of cyclic and negacyclic codes of length <inline-formula><tex-math id="M14">\begin{document}$ 8p^s $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M15">\begin{document}$ \mathcal R $\end{document}</tex-math></inline-formula>.</p>

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