Abstract
In this article, by using Hermitian construction, we obtain several new quantum codes and quantum MDS codes compare to the known codes from the constacyclic codes with only one q2-cyclotomic coset containing at least two consecutive integers with arbitrary difference r but fixed. This work is the generalization of the recent study on quantum codes from cyclic codes by La Guardia (Int. J. Theor. Phys. 56(8): 2479–2484, 2017) and from negacyclic codes by Gao and Wang (Int. J. Theor. Phys. 57(3): 682–686, 2018), respectively.
Similar content being viewed by others
References
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. 11(2), 325–335 (2019)
Calderbank, A.R., Rains, E.M., Shor, P.M., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inform. Theory. 44, 1369–1387 (1998)
Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inform. Theory. 61, 1474–1484 (2015)
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(1-8) (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. 17(1), 9 (2018)
Gao, J., Wang, Y.: Quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 57(3), 682–686 (2018)
Grassl, M., Beth, T.: On optimal quantum codes. Int. J. Quantum Inf. 2, 55–64 (2004)
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(1–2), 625–635 (2019)
Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inform. Theory. 60(4), 2080–2085 (2014)
Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inform. Theory. 52(11), 4892–4914 (2006)
La Guardia, G.G., Palazzo, R. Jr: Constructions of new families of nonbinary CSS codes. Discrete Math. 310, 2935–2945 (2010)
La Guardia, G.G.: New quantum MDS codes. IEEE Trans. Inform. Theory. 57 (8), 5551–5554 (2011)
La Guardia, G.G.: On the construction of nonbinary quantum BCH codes. IEEE Trans. Inform. Theory. 60(3), 1528–1535 (2014)
La Guardia, G.G.: Quantum codes derived from cyclic codes. Int. J. Theor. Phys. 56(8), 2479–2484 (2017)
Liu, Y., Li, R., Lv, L., Ma, M.: A class of constacyclic BCH codes and new quantum codes. Quantum Inf Process. 16(3), 16 (2017)
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(6), 19 (2018)
Qian, J., Ma, W., Gou, W.: Quantum codes from cyclic codes over finite ring. Int. J. Quantum Inf. 7, 1277–1283 (2009)
Shor, P.W.: Scheme for reducing decoherence in quantum memory. Phys. Rev. A. 52, 2493–2496 (1995)
Taneja, D., Gupta, M., Narula, R., Bhullar, J.: Construction of new quantum MDS codes derived from constacyclic codes. Int. J. Quantum Inf. 15(1), 1750008 (2017)
Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf Process. 14(3), 881–889 (2015)
Yang, Y., Cai, W.: On self-dual constacyclic codes over finite fields. Des. Codes Cryptogr. 74(2), 355–364 (2013)
Acknowledgments
The authors are thankful to the University Grants Commission (UGC), Govt. of India for financial support and Indian Institute of Technology Patna for providing research facilities. The authors would like to thank the anonymous referee(s) for their careful reading and valuable comments, which helped to improve the presentation of the manuscript.
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. & Bhunia, D.K. Quantum Codes Obtained from Constacyclic Codes. Int J Theor Phys 58, 3945–3951 (2019). https://doi.org/10.1007/s10773-019-04260-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-019-04260-y