Abstract
Let \({\mathbb {Z}}_{4}\) be the ring of integers modulo 4. This paper presents \((1+2u+2v+2uv)\)-constacyclic and skew \((1+2u+2v+2uv)\)-constacyclic codes over the ring \( {\mathbb {Z}}_{4} +u{\mathbb {Z}}_{4}+v{\mathbb {Z}}_{4}+uv{\mathbb {Z}}_{4} \) where \(u^2=u,v^{2}=v, uv=vu\). We define three Gray maps and show that the Gray images of \((1+2u+2v+2uv)\)-constacyclic and skew \((1+2u+2v+2uv)\)-constacyclic codes are cyclic, quasi-cyclic and permutation equivalent to quasi-cyclic codes over \({\mathbb {Z}}_4\). Also, we show that cyclic and \((1+2u+2v+2uv)\)-constacyclic codes of odd length are principally generated. As an application, several new quaternary linear codes from the Gray images of \((1+2u+2v+2uv)\)-constacyclic codes are obtained.
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Acknowledgements
The authors are thankful to the University Grant Commission(UGC), Govt. of India for financial support, and Indian Institute of Technology Patna for providing the research facilities. Authors are also thankful to Prof. Patrick Sol\(\acute{e},\) (CNRS, Aix-Marseille University, Centrale Marseille), France for his constructive comments to improve the results and presentation of the manuscript. Furthermore, the authors would like to thank the anonymous referee(s) for their careful reading and valuable comments.
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Communicated by Thomas Aaron Gulliver.
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Islam, H., Prakash, O. New \({\mathbb {Z}}_4\) codes from constacyclic codes over a non-chain ring. Comp. Appl. Math. 40, 12 (2021). https://doi.org/10.1007/s40314-020-01398-y
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DOI: https://doi.org/10.1007/s40314-020-01398-y