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Optimal quinary cyclic codes with three zeros

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Abstract

Cyclic codes are an important subclass of linear codes, they not only have good algebraic structure, but also are easy to be encoded and decoded. At present, researchers have constructed many optimal ternary cyclic codes, but the study on quinary cyclic codes is less developed. In this paper, by analyzing the solutions of certain equations over \(\mathbb {F}_{5^m}\), we construct some optimal quinary cyclic codes with three zeros and with parameters \([5^m-1, 5^m-2-2m, 4]\), \([5^m-1, 5^m-2-\frac{3m}{2}, 4]\). Moreover, the weight distributions of two classes of their duals are also provided.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China(No. U21A20428, NO. 12171134, NO. 62201009, NO. 12201170) and the Natural Science Foundation of Anhui Province(NO. 2108085QA06, NO. 2108085QA03).

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Authors and Affiliations

  1. School of Mathematics, HeFei University of Technology, 230009, Hefei, China

    Tingting Wu, Shixin Zhu & Li Liu

  2. Intelligent Interconnected Systems Laboratory of Anhui Province, Hefei, China

    Tingting Wu, Shixin Zhu & Li Liu

  3. School of Information and Artificial Intelligence, Anhui Agricultural University, 230036, Hefei, Anhui, China

    Lanqiang Li

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  1. Tingting Wu
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  2. Shixin Zhu
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Contributions

Tingting Wu and Shixin Zhu wrote the main manuscript text; Li Liu and Lanqiang Li collected a lot of information related to our paper; All authors reviewed the manuscript.

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Correspondence to Shixin Zhu.

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Wu, T., Zhu, S., Liu, L. et al. Optimal quinary cyclic codes with three zeros. Cryptogr. Commun. 16, 801–823 (2024). https://doi.org/10.1007/s12095-024-00703-4

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Keywords

Mathematics Subject Classification (2010)