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Quantum codes from codes over the ring \({\pmb {\mathbb {F}}}_{q}+\alpha \pmb {\mathbb {F}}_{q}\)

  • Murat GüzeltepeEmail author
  • Mustafa Sarı
Article
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Abstract

In this paper, we aim to obtain quantum error correcting codes from codes over a nonlocal ring \(R_q={\mathbb {F}}_q+\alpha {\mathbb {F}}_q\). We first define a Gray map \(\varphi \) from \(R_q^n\) to \({\mathbb {F}}_q^{2n}\) preserving the Hermitian orthogonality in \(R_q^n\) to both the Euclidean and trace-symplectic orthogonality in \({\mathbb {F}}_q^{2n}\). We characterize the structure of cyclic codes and their duals over \(R_q\) and derive the condition of existence for cyclic codes containing their duals over \(R_q\). By making use of the Gray map \(\varphi \), we obtain two classes of q-ary quantum codes. We also determine the structure of additive cyclic codes over \(R_{p^2}\) and give a condition for these codes to be self-orthogonal with respect to Hermitian inner product. By defining and making use of a new map \(\delta \), we construct a family of p-ary quantum codes.

Keywords

Quantum codes Cyclic codes Gray map 

Notes

Acknowledgements

The work was supported by TÜBİTAK (The Scientific and Technological Research Council of TURKEY) with Project Number 116F318. The author wishes to thank the associate editor and the anonymous referee whose comments have greatly improved this paper.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsSakarya UniversitySakaryaTurkey
  2. 2.Department of MathematicsYıldız Technical UniversityIstanbulTurkey

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