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Designs, Codes and Cryptography

, Volume 87, Issue 1, pp 15–29 | Cite as

Several classes of linear codes with a few weights from defining sets over \(\mathbb {F}_p+u\mathbb {F}_p\)

  • Haibo Liu
  • Qunying LiaoEmail author
Article
  • 138 Downloads

Abstract

Recently, linear codes with a few weights have been extensively studied due to their applications in secret sharing schemes, authentication codes, constant composition codes. Results have shown that some optimal codes can be acquired if the defining sets are well chosen over finite fields. In this paper, we investigate the Lee-weight distribution of linear codes over the ring \(\mathbb {F}_p +u\mathbb {F}_p\) (p is an odd prime) based on defining sets by employing exponential sums. We then determine the explicit complete weight enumerator for the images of these linear codes under the Gray map. A class of constant weight codes that meets the Griesmer bound for constructing optimal constant composition codes achieving the LVFC bound is also presented.

Keywords

Exponential sum Linear code Weight distribution Complete weight enumerator Constant composition code 

Mathematics Subject Classification

94B05 94B15 94B60 

Notes

Acknowledgements

The authors wish to thank the reviewers for valuable comments and suggestions which greatly helped us to improve this paper. This work is supported by the Natural Science Foundation of China with No. 11401408, Project of science and Technology Department of Sichuan Province with No. 2016JY0134.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and Software ScienceSichuan Normal UniversityChengduChina

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