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Complete classification of \((\delta +\alpha u^2)\)-constacyclic codes over \({\mathbb {F}}_{3^m}[u]/\langle u^4\rangle \) of length 3n

  • Yuan Cao
  • Yonglin CaoEmail author
  • Li Dong
Original Paper

Abstract

Let \({\mathbb {F}}_{3^m}\) be a finite field of cardinality \(3^m\), \(R={\mathbb {F}}_{3^m}[u]/\langle u^4\rangle \) which is a finite chain ring, and n be a positive integer satisfying \(\mathrm{gcd}(3,n)=1\). For any \(\delta ,\alpha \in {\mathbb {F}}_{3^m}^{\times }\), an explicit representation for all distinct \((\delta +\alpha u^2)\)-constacyclic codes over R of length 3n is given, formulas for the number of all such codes and the number of codewords in each code are provided, respectively. Moreover, the dual code for each of these codes is determined explicitly.

Keywords

Constacyclic code Dual code Linear code Finite chain ring 

Mathematics Subject Classification

94B15 94B015 11T71 

Notes

Acknowledgements

We thank the anonymous referees for valuable comments that improved the presentation of this paper. Part of this work was done when Yonglin Cao was visiting Chern Institute of Mathematics, Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality. This research is supported in part by the National Natural Science Foundation of China (Grant Nos. 11671235, 11471255).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of ScienceShandong University of TechnologyZiboChina

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