Cyclic codes over \({\mathcal {M}}_4({\mathbb {F}}_2\))

  • Joydeb PalEmail author
  • Sanjit Bhowmick
  • Satya Bagchi
Original Research


In this article, keeping the huge research prospective of the study in mind, we consider the non-commutative ring \({\mathcal {M}}_4({\mathbb {F}}_2)\), the set of all \(4 \times 4\) matrices over the field \({\mathbb {F}}_2\) and confirm that this ring is isomorphic with the ring \({\mathbb {F}}_{16}+u {\mathbb {F}}_{16}+u^2 {\mathbb {F}}_{16}+u^3{\mathbb {F}}_{16}\), where \(u^4=0\). Besides, we develop the structure of cyclic codes and their generators over the ring. Also, making use of Gray map from \({\mathcal {M}}_4({\mathbb {F}}_2)\) to \({\mathbb {F}}_{16}^4\), we infer that the image of a cyclic code is a linear code. Finally, our findings are authenticated by suitable non-trivial examples.


Matrix ring Modules Cyclic codes Quaternary codes 

Mathematics Subject Classification

94B05 94B15 



The authors are thankful to the editor and the anonymous referees for their valuable comments and suggestions. The first named author Joydeb Pal would like to convey cordial thanks to DST-INSPIRE, and the second named author Sanjit Bhowmick is thankful to MHRD for financial supports to pursue their research works. This work was also supported by DST-SERB, India (Grant No. EEQ/2016/000140).


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Copyright information

© Korean Society for Computational and Applied Mathematics 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology DurgapurBurdwanIndia

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