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On some conjectures about optimal ternary cyclic codes

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Abstract

Cyclic codes are a subclass of linear codes and have efficient encoding and decoding algorithms over finite fields, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, by considering the solutions of certain equations over finite fields, one of the nine conjectures proposed by Ding and Helleseth about optimal cyclic codes in (IEEE Trans Inf Theory 59(9):5898–5904, 2013) is settled. In addition, we make progress toward other two conjectures.

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Acknowledgements

The authors are very grateful to the three anonymous reviewers, the Editor in Chief and CE, Prof. Tor Helleseth, for their comments which improved the presentation and quality of this paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Yan Liu & Xiwang Cao

  2. College of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, 224003, People’s Republic of China

    Yan Liu

  3. School of Mathematics, Southeast University, Nanjing, 211189, People’s Republic of China

    Wei Lu

Authors
  1. Yan Liu
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  2. Xiwang Cao
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  3. Wei Lu
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Corresponding author

Correspondence to Xiwang Cao.

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Communicated by T. Helleseth.

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Y. Liu: is supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 19KJB120014). X. Cao: is supported by the National Natural Science Foundation of China (No. 11771007). W. Lu: is supported by the National Natural Science Foundation of China (No. 11801070).

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Liu, Y., Cao, X. & Lu, W. On some conjectures about optimal ternary cyclic codes. Des. Codes Cryptogr. 88, 297–309 (2020). https://doi.org/10.1007/s10623-019-00679-w

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  • DOI: https://doi.org/10.1007/s10623-019-00679-w

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