Abstract
In this article, for any odd prime p, we construct the quantum codes over \(\mathbb {F}_{p}\) by using the cyclic codes of length n over \(R=\mathbb {F}_{p}[u,v,w]/\langle u^{2}-1,v^{2}-1,w^{2}-1,uv-vu,vw-wv,wu-uw\rangle \). We obtain the self-orthogonal properties of cyclic codes over R and as an application, present some new quantum codes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over \(\mathbb{F}_{p} +v\mathbb{F}_{p}\). Int. J. Inf. Coding Theory 3(2), 137–144 (2015)
Ashraf, M., Mohammad, G.: Quantum codes from cyclic codes over \(\mathbb{F}_{q} +u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}\). Quantum Inf. Process. 15(10), 4089–4098 (2016)
Ashraf, M., Mohammad, G.: Quantum codes over \(\mathbb{F}_{p}\) from cyclic codes over \(\mathbb{F}_{p}[u,v]/\langle u^{2}-1,v^{3}-v,uv-vu\rangle \). Cryptogr. Commun. (2018) https://doi.org/10.1007/s12095-018-0299-0
Bosma, W., Cannon, J.: Handbook of Magma Functions. Univ. of Sydney, Sydney (1995)
Calderbank, A.R., Rains, E.M., Shor, P.M., Sloane, N.J.A.: Quantum error correction via codes over \(GF(4)\). IEEE Trans. Inf. Theory 44, 1369–1387 (1998)
Dertli, A., Cengellenmis, Y., Eren, S.: On quantum codes obtained from cyclic codes over \(A_2\). Int. J. Quantum Inf. 13(3), 1550031 (2015)
Gao, J.: Quantum codes from cyclic codes over \(\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}+v^{3}\mathbb{F}_{q}\). Int. J. Quantum Inf. 13(8), 1550063(1)–1550063(8) (2015)
Gao, J.: Some results on linear codes over \(\mathbb{F}_{p} +u\mathbb{F}_{p}+u^{2}\mathbb{F}_{p}\). J. Appl. Math. Comput. 47, 473–485 (2015)
Gao, J., Wang, Y.: \(u\)-Constacyclic codes over \(\mathbb{F}_{p}+u\mathbb{F}_{p}\) and their applications of constructing new non-binary quantum codes. Quantum Inf. Process. (2018) https://doi.org/10.1007/s11128-017-1775-8
Gaurdia, G., Palazzo Jr., R.: Constructions of new families of nonbinary CSS codes. Discrete Math. 310, 2935–2945 (2010)
Gottesman, D.: An introduction to quantum error-correction. Proc. Symp. Appl. Math. 68, 13–27 (2010)
Grassl, M., Beth, T.: On optimal quantum codes. Int. J. Quantum Inf. 2, 55–64 (2004)
Islam, H., Verma, R.K., Prakash, O.: A family of constacyclic codes over \(\mathbb{F}_{p^m}[v, w]/\langle v^{2}-1, w^{2}-1, vw-wv\rangle \). Int. J. Inf. Coding Theory (2018). (in press)
Kai, X., Zhu, S.: Quaternary construction of quantum codes from cyclic codes over \(\mathbb{F}_{4}+u\mathbb{F}_{4}\) Int. J. Quantum Inf. 9, 689–700 (2011)
Li, R., Xu, Z., Li, X.: Binary construction of quantum codes of minimum distance three and four. IEEE Trans. Inf. Theory 50, 1331–1335 (2004)
Li, J., Gao, J., Wang, Y.: Quantum codes from \((1-2v)\)-constacyclic codes over the ring \(\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}\). Discrete Math. Algorithms Appl. 10(4), 1850046 (2018)
Ma, F., Gao, J., Fu, F.W.: Constacyclic codes over the ring \(\mathbb{F}_{p} +v\mathbb{F}_{p}+v^{2}\mathbb{F}_{p}\) and their applications of constructing new non-binary quantum codes. Quantum Inf. Process. 17, 122 (2018). https://doi.org/10.1007/s11128-018-1898-6
Ozen, M., Ozzaim, N.T., Ince, H.: Quantum codes from cyclic codes over \(\mathbb{F}_{3} +u\mathbb{F}_{3}+v\mathbb{F}_{3}+uv\mathbb{F}_{3}\). Int. Conf. Quantum Sci. Appl. J. Phys. Conf. Ser. 766, 012020-1–012020-6 (2016)
Qian, J., Ma, W., Gou, W.: Quantum codes from cyclic codes over finite ring. Int. J. Quantum Inf. 7, 1277–1283 (2009)
Singh, A.K., Pattanayek, S., Kumar, P.: On quantum codes from cyclic codes over \(\mathbb{F}_{2} +u\mathbb{F}_{2}+u^2\mathbb{F}_{2}\). Asian Eur. J. Math. 11(1), 1850009 (2018)
Shor, P.W.: Scheme for reducing decoherence in quantum memory. Phys. Rev. A 52, 2493–2496 (1995)
Acknowledgements
The authors are thankful to the University Grants Commission (UGC) for financial support and Indian Institute of Technology Patna for providing research facilities. Also, the authors would like to thank the anonymous referee(s) and the editor for their valuable comments to improve the presentation of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Islam, H., Prakash, O. Quantum codes from the cyclic codes over \(\mathbb {F}_{p}[u,v,w]/\langle u^{2}-1,v^{2}-1,w^{2}-1,uv-vu,vw-wv,wu-uw\rangle \). J. Appl. Math. Comput. 60, 625–635 (2019). https://doi.org/10.1007/s12190-018-01230-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-018-01230-1