Cryptography and Communications

, Volume 10, Issue 6, pp 1109–1117 | Cite as

Cyclic codes over \(M_{2}(\mathbb {F}_{2}+u\mathbb {F}_{2})\)

  • Rong LuoEmail author
  • Udaya Parampalli
Part of the following topical collections:
  1. Special Issue on Sequences and Their Applications


Let \(A=M_{2}(\mathbb {F}_{2}+u\mathbb {F}_{2})\), where u 2 = 0, the ring of 2 × 2 matrices over the finite ring \(\mathbb {F}_{2}+u\mathbb {F}_{2}\). The ring A is a non-commutative Frobenius ring but not a chain ring. In this paper, we derive the structure theorem of cyclic codes of odd length over the ring A and use them to construct some optimal cyclic codes over \(\mathbb {F}_{4}\). Let v 2 = 0 and u v = v u. We also give an isometric map from A to \(\mathbb {F}_{4}+v\mathbb {F}_{4}+u\mathbb {F}_{4}+uv\mathbb {F}_{4}\) using their respective Bachoc weight and Lee weight.


Cyclic codes Lee weight Bachoc weight Gray map 

Mathematics Subject Classification (2010)

94B05 94B15 



The authors would like to thank the Editor and anonymous reviewers for their valuable suggestions and comments that have much improved the quality of this paper. This research is supported in part by the National Natural Science Foundation of China Under Grants 11401488.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of MathematicsSouthwest Jiaotong UniversityChengduChina
  2. 2.Department of Computing and Information SystemsUniversity of MelbourneVictoriaAustralia

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