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Self-dual cyclic codes over \({\mathbb {Z}}_4\) of length 4n

Abstract

For any odd positive integer n, we express cyclic codes over \({\mathbb {Z}}_4\) of length 4n in a new way. Based on the expression of each cyclic code \({\mathcal {C}}\), we provide an efficient encoder and determine the type of \({\mathcal {C}}\). In particular, we give an explicit representation and enumeration for all distinct self-dual cyclic codes over \({\mathbb {Z}}_4\) of length 4n and correct a mistake in the paper “Concatenated structure of cyclic codes over \({\mathbb {Z}}_4\) of length 4n” (Cao et al. in Appl Algebra Eng Commun Comput 10:279–302, 2016). In addition, we obtain 50 new self-dual cyclic codes over \({\mathbb {Z}}_4\) of length 28.

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Acknowledgements

This research is supported in part by the National Natural Science Foundation of China (Grant Nos. 11671235, 11801324, 61971243, 61571243), the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2018BA007) and the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University) (Grant Nos. HBAM201906, HBAM201804), the Scientific Research Fund of Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2018MMAEZD09) and the Nankai Zhide Foundation. Part of this work was done when Yonglin Cao was visiting Chern Institute of Mathematics, Nankai University, Tianjin, China. He would like to thank the institution for the kind hospitality.

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Correspondence to Yonglin Cao.

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Cao, Y., Cao, Y., Fu, FW. et al. Self-dual cyclic codes over \({\mathbb {Z}}_4\) of length 4n. AAECC 33, 21–51 (2022). https://doi.org/10.1007/s00200-020-00424-0

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  • DOI: https://doi.org/10.1007/s00200-020-00424-0

Keywords

  • Self-dual code
  • Cyclic code
  • Encoder
  • Galois ring

Mathematics Subject Classification

  • 94B15
  • 94B05
  • 11T71