Abstract
In this paper, let \(\alpha +u\beta \) be a unit in \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) \((u^2=0),\) where p is an odd prime, m is a positive integer and \(\beta \ne 0\). With the help of decomposition of the binomial \(x^{8}- \alpha _0\) into a product of irreducible coprime polynomials and the ring \(\frac{({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m})[x]}{\left\langle x^{8p^s}-(\alpha +u\beta )\right\rangle }\) is a principal ideal ring, we give the complete description of all \((\alpha +u\beta )\)-constacyclic codes of length \(8p^s\) over the finite commutative chain ring \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) \((u^2=0)\) in terms of their generator polynomials, where \(\alpha _{0}^{p^{s}}=\alpha \). We also find out the number of codewords in each of these constacyclic codes. Besides illustrating our results with examples, we determine duals of constacyclic codes, and as an application, we determine the self-dual, self-orthogonal, dual-containing, and linear complimentary-dual \((\alpha +u\beta )\)-constacyclic codes of length \(8p^s\) over \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\) \((u^2=0)\). Also, we determine the RT (Rosenbloom–Tsfasman) distances, RT weight distributions, and Hamming distances of such constacyclic codes.
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Acknowledgements
The authors are greatful to Dr. Abhishake Rastogi for his useful suggestions and ideas which helped the authors to drive and improve the results. The authors would like to sincerely thank the referees for a very meticulous reading of this manuscript, and for valuable suggestions which helped to create an improved final version. The work of H.Q. Dinh is supported in part by the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University. The work of Saroj Rani is supported by the Department of Science and Technology (DST), India under the Grant No. MTR/2018/001250.
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Communicated by Thomas Aaron Gulliver.
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Rani, S., Dinh, H.Q. RT distances and Hamming distances of constacyclic codes of length \(8p^s\) over \({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}\). Comp. Appl. Math. 41, 159 (2022). https://doi.org/10.1007/s40314-022-01867-6
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DOI: https://doi.org/10.1007/s40314-022-01867-6
Keywords
- Constacyclic codes
- Dual codes
- Self-orthogonal codes
- Self-dual codes
- Dual-containing codes
- RT distance
- Hamming distance