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On the construction of self-dual cyclic codes over \(\mathbb {Z}_{4}\) with arbitrary even length

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Abstract

Self-dual codes over the ring \(\mathbb {Z}_{4}\) are related to combinatorial designs and unimodular lattices. First, we discuss briefly how to construct self-dual cyclic codes over \(\mathbb {Z}_{4}\) of arbitrary even length. Then we focus on solving one key problem of this subject: for any positive integers k and m such that m is even, we give a direct and effective method to construct all distinct Hermitian self-dual cyclic codes of length 2k over the Galois ring GR(4,m). This then allows us to provide explicit expressions to accurately represent all these Hermitian self-dual cyclic codes in terms of binomial coefficients. In particular, several numerical examples are presented to illustrate our applications.

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Acknowledgements

This research is supported in part by the National Natural Science Foundation of China (Grant Nos. 12071264, 11801324, 11671235), the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2018BA007), Nanyang Technological University, Singapore (Grant No. 04INS000047C230GRT01), the IC Program of Shandong Institutions of Higher Learning For Youth Innovative Talents and the Scientific Research Fund of Hubei Provincial Key Laboratory of Applied Mathematics (Hubei University)(Grant Nos. HBAM201906).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong, 255091, China

    Yuan Cao, Yonglin Cao & Guidong Wang

  2. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China

    Yuan Cao

  3. School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Republic of Singapore

    San Ling

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  1. Yuan Cao
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Correspondence to Yonglin Cao.

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Cao, Y., Cao, Y., Ling, S. et al. On the construction of self-dual cyclic codes over \(\mathbb {Z}_{4}\) with arbitrary even length. Cryptogr. Commun. 14, 1117–1143 (2022). https://doi.org/10.1007/s12095-022-00579-2

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