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Constacyclic codes over mixed alphabets and their applications in constructing new quantum codes

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Let p be an odd prime and m be a positive integer, \(q=p^m\), \({\mathcal {R}}={\mathbb {F}}_q+u{\mathbb {F}}_q\) with \(u^2=u\), and \({\mathcal {S}}={\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q\) with \(u^2=u, v^2=v, uv=vu=0\) and \(\Lambda =(\lambda _1,\lambda _2,\lambda _3)\in {\mathbb {F}}_q{\mathcal {R}}{\mathcal {S}}\). In this paper, we study the algebraic structure of constacyclic codes over \({\mathcal {R}}\) and \({\mathcal {S}}\). Further, we discuss the structure of \({\mathbb {F}}_q{\mathcal {R}}{\mathcal {S}}\)-\(\Lambda \)-constacyclic codes of block length \((\alpha ,\beta ,\gamma )\). This family of codes can be viewed as \({\mathcal {S}}[x]\)-submodules of \(\frac{{\mathbb {F}}_q[x]}{\langle x^{\alpha }-\lambda _1\rangle }\times \frac{{\mathcal {R}}[x]}{\langle x^{\beta }-\lambda _2\rangle }\times \frac{{\mathcal {S}}[x]}{\langle x^{\gamma }-\lambda _3\rangle }\). The generator polynomials of this family of codes are discussed. As application, we discuss the construction of quantum error-correcting codes (QECCs) from constacyclic codes over \({\mathbb {F}}_q{\mathcal {R}}{\mathcal {S}}\) and obtain several new QECCs from this study.

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Acknowledgements

The authors are grateful to the anonymous reviewers who have given us very thoughtful and helpful comments to improve the manuscript. The second and third authors are thankful to University Grant Commission (UGC), Govt. of India, for financial support. The authors are grateful to the Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, for partial financial support. AKU thanks SERB DST for its support through Project No. MTR/2020/000006 under MATRICS scheme.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Warren, OH, 44483, USA

    Hai Q. Dinh

  2. Department of Mathematics, Indian Institute of Technology Patna, Patna, 801103, India

    Sachin Pathak & Tushar Bag

  3. Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, 221005, India

    Ashish Kumar Upadhyay

  4. Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Chiang Mai, 50200, Thailand

    Hai Q. Dinh & Woraphon Yamaka

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  1. Hai Q. Dinh
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Correspondence to Sachin Pathak.

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Dinh, H.Q., Pathak, S., Bag, T. et al. Constacyclic codes over mixed alphabets and their applications in constructing new quantum codes. Quantum Inf Process 20, 150 (2021). https://doi.org/10.1007/s11128-021-03083-3

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