On some conjectures about optimal ternary cyclic codes

  • Yan Liu
  • Xiwang CaoEmail author
  • Wei Lu


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.


Cyclic code Optimal code Sphere packing bound 

Mathematics Subject Classification

94B15 11T71 



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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Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  2. 2.College of Mathematics and PhysicsYancheng Institute of TechnologyYanchengPeople’s Republic of China
  3. 3.School of MathematicsSoutheast UniversityNanjingPeople’s Republic of China

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