Abstract
For an odd prime p, this article studies the \(\lambda \)-constacyclic and skew \(\lambda \)-constacyclic codes of arbitrary length over the finite commutative non-chain ring \(R=\mathbb {F}_{p^m}[u,v,w]/\langle u^{2}-1,v^{2}-1,w^{2}-1,uv-vu,vw-wv,wu-uw\rangle \), where \(\lambda \) is a unit in R. By using the decomposition method, we determine the structure of \(\lambda \)-constacyclic and skew \(\lambda \)-constacyclic codes. Also, the necessary and sufficient conditions of these codes to be self-dual are obtained. Further, it is shown that the Gray images of \(\lambda \)-constacyclic and skew \(\lambda \)-constacyclic codes of length n over R are quasi-twisted and skew quasi-twisted codes, respectively of length 8n and index 8 over \(\mathbb {F}_{p^m}\). Finally, two non-trivial examples are given to validate the obtained results.
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Acknowledgements
The authors are thankful to the University Grants Commission (UGC) (under Sr. No. 2121540952, Ref. No. 20/12/2015(ii)EU-V dated 31/08/2016) and the Council of Scientific & Industrial Research (CSIR) (under grant no. 09/1023(0014)/2015-EMR-I), Govt. of India for financial supports and Indian Institute of Technology Patna for providing research facilities.
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Prakash, O., Islam, H., Krishna Verma, R. (2023). Constacyclic and Skew Constacyclic Codes Over a Finite Commutative Non-chain Ring. In: Silvestrov, S., Malyarenko, A. (eds) Non-commutative and Non-associative Algebra and Analysis Structures. SPAS 2019. Springer Proceedings in Mathematics & Statistics, vol 426. Springer, Cham. https://doi.org/10.1007/978-3-031-32009-5_26
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