Abstract
Let q be an odd prime power, and w be a generator of the multiplicative cyclic group \({\mathbb {F}}^*_{q^2} \). In this work, we study constacyclic codes over \(R={\mathbb {F}}_{q^2} +u{\mathbb {F}}_{q^2}~ (u^2=w^2)\), and show a construction of quantum codes by studying the Hermitian construction over R. To do that, we discuss the \(\sigma \)-inner product over any finite commutative Frobenius ring and using that we define the Hermitian inner product over R. We decompose the ring R by constructing a pairwise orthogonal idempotent and study linear codes. We present the units of R and using them we study constacyclic codes and their generators over R. We also provide a condition for a constacyclic code to contain its Hermitian dual over this ring. Finally, we constructed some codes, which have improved parameters than some of the recently constructed codes in the literature. We also present a table of codes, which have better parameters than those currently available in the online database.
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Acknowledgements
The authors would like to sincerely thank the referees for a very meticulous reading of this manuscript, and for valuable suggestions which help to create an improved final version. T. Bag and K. Abdukhalikov are supported by the UAEU Grant G00003491. A. K. Upadhyay is grateful to SERB DST Govt of India for financial support under MATRICS scheme with file number MTR/2020/000006. H. Q. Dinh and W. Yamaka are thankful for the partially supported by the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Thailand.
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Bag, T., Dinh, H.Q., Abdukhalikov, K. et al. Constacyclic codes over \({\pmb {\mathbb {F}}}_{q^2}[u]/\langle u^2-w^2 \rangle \) and their application in quantum code construction. J. Appl. Math. Comput. 68, 3821–3834 (2022). https://doi.org/10.1007/s12190-021-01692-w
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DOI: https://doi.org/10.1007/s12190-021-01692-w