Abstract
Let \(R=\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \). Then R is a local non-principal ideal ring of 16 elements. First, we give the structure of every cyclic code of odd length n over R and obtain a complete classification for these codes. Then we determine the cardinality, the type and its dual code for each of these cyclic codes. Moreover, we determine all self-dual cyclic codes of odd length n over R and provide a clear formula to count the number of these self-dual cyclic codes. Finally, we list some optimal 2-quasi-cyclic self-dual linear codes of length 30 over \(\mathbb {Z}_{4}\) and obtain 4-quasi-cyclic and formally self-dual binary linear [60,30,12] codes derived from cyclic codes of length 15 over \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \).
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Acknowledgements
Part of this work was done when Yonglin Cao was visiting the Chern Institute of Mathematics, Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality. This research is supported in part by the National Natural Science Foundation of China (Grant Nos. 11671235, 11801324), the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2018BA007), the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University)(Grant No. AM201804) and the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD09). Yuan Cao and Yonglin Cao contribute equally to this article.
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Cao, Y., Cao, Y. Complete classification for simple root cyclic codes over the local ring \(\mathbb {Z}_{4}[v]/\langle v^{2}+2v\rangle \). Cryptogr. Commun. 12, 301–319 (2020). https://doi.org/10.1007/s12095-019-00403-4
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DOI: https://doi.org/10.1007/s12095-019-00403-4