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Designs, Codes and Cryptography

, Volume 75, Issue 2, pp 263–275 | Cite as

The weight distribution of a family of \(p\)-ary cyclic codes

  • Dabin ZhengEmail author
  • Xiaoqiang Wang
  • Xiangyong Zeng
  • Lei Hu
Article

Abstract

Let \(p\) be an odd prime, and \(m\) and \(k\) be two positive integers with \(\frac{m}{\gcd (m,k)}\) being odd. This paper determines the weight distribution of a family of \(p\)-ary cyclic codes over \({\mathbb {F}}_p\) whose duals have three zeros \(\alpha ^{-2}, \alpha ^{-(p^{2k}+1)}\) and \(\alpha ^{-(p^{4k}+1)}\), where \(\alpha \) is a primitive element of \({\mathbb {F}}_{p^m}\).

Keywords

Cyclic code Weight distribution Exponential sum  Quadratic form 

Mathematics Subject Classification

11T71 94B15 

Notes

Acknowledgments

The authors wish to thank anonymous referees for their helpful comments, which have improved the presentation of this paper. The work was partially supported by National Natural Science Foundation of China under Grants 11101131, 61170257, 10990011 and the National Basic Research Program of China (2013CB834203).

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Dabin Zheng
    • 1
    Email author
  • Xiaoqiang Wang
    • 1
  • Xiangyong Zeng
    • 1
    • 2
  • Lei Hu
    • 2
    • 3
  1. 1.Faculty of Mathematics and StatisticsHubei UniversityWuhanChina
  2. 2.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina
  3. 3.Beijing Center for Mathematics and Information Interdisciplinary SciencesBeijingChina

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