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Quantum error-correcting codes from the quantum construction X

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Abstract

In this paper, we first modify a result of constructing quantum error-correcting (QEC) codes via Hermitian dual to via Euclidean dual over finite fields. Then, we give five methods of constructing QEC codes. In addition, we construct QEC codes to have better parameters than the ones available in the literature.

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References

  1. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183–1188 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bag, T., Dinh, H. Q., Abdukhalikov, K., Upadhyay, A. k., Yamaka, W.: Constacyclic codes over \(\mathbb{ F} _{q^2}/ \langle u^2-w^2\rangle \) and their application in quantum code construction. J. Appl. Math. Comput. 68, 3821–3834 (2022)

  4. Bag, T., Dinh, H.Q., Upadhyay, A.K., Bandi, R., Yamaka, W.: Quantum codes from skew constacyclic codes over the ring \(\mathbb{F} _p[u, v, w]/\langle u^2-1, v^2- 1, uv-vu\rangle \). Discrete Math. 343, 111737 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blackmore, T., Norton, G. H.: Matrix-product codes over \(\mathbb{F}_q\). Appl. Algebra Eng. Commun. Comput. 12, 477–500 (2001)

  6. Bll, S.: Some constructions of quantum MDS codes. Des. Codes Cryptogr. 73(2), 417–424 (2020)

  7. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44(4), 1369–1387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Castagnoli, G., Massey, J. L., Schoeller, P. A., Seemann, N. V.: On repeated-root cyclic codes. IEEE Trans. Inf. Theory. 37(2), 337–342 (1991)

  9. Degwekar, A., Guenda, K., Gulliver, T. A.: Extending construction X for quantum error-correcting codes. In: Coding Theory and Applications, pp. 141–152. Springer (2015)

  10. Ding, J., Li, H., Liang, J., Tang, Y.: Quantum codes from constacyclic codes over polynomial residue rings. Chin. J. Electron. 28(6), 1131–1138 (2019)

    Article  Google Scholar 

  11. Dinh, H. Q., Bag, T., Abdukhalikov, K., Pathak, S., Bandi, R., Chinnakum, W.: On a class of skew constacyclic codes over mixed alphabets and applications in constructing optimal and quantum codes. Cryptogr. Commun. https://doi.org/10.1007/s12095-022-00594-3

  12. Edel, Y.: Some good quantum twisted codes. https://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/QTBCHIndex.html. Accessed July, 2020

  13. Ezerman, M. F., Ling, S., Özkaya, B., Solè, P.: Good stabilizer codes from quasi-cyclic codes over \(\mathbb{F} _4\) and \(9\). In: IEEE International Symposium on Information Theory, pp. 2898–2902 (2019)

  14. Fang, W., Fu, F.: Some new constructions of quantum MDS codes. IEEE Trans. Inf. Theory 65(12), 7840–7847 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  15. Feng, K.: Algebraic Theory of Error-Correcting Codes. Tsinghua University Press, Beijing (2005)

    Google Scholar 

  16. Gao, Y., Gao, J., Fu, F.: Quantum codes from cyclic codes over the ring \(\mathbb{F} _q +v_1\mathbb{ F} _q +\cdots + v_r\mathbb{ F} _q\). Appl. Algebra Eng. Commun. Comput. 30, 161–174 (2019)

    Article  Google Scholar 

  17. Grassl, M., Beth, T., Röttler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2(1), 757–775 (2004)

    Article  Google Scholar 

  18. Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de. Accessed 30-2021 (2007)

  19. Guenda, K, Gulliver T. A., Jitman, S., Thipworawimon. S.: Linear \(l\)-intersection pairs of codes and their applications. Des. Codes Cryptogr. 88, 133–152 (2020)

  20. Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  21. Jin, L., Ling, S., Luo, J.: Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes. IEEE Trans. Inf. Theory 56(9), 4735–4740 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jin, L., Xing, C.: Construction of MDS codes with complementary duals. IEEE Trans. Inf. Theory 63, 2843–2847 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193–1197 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory. 52(11), 4892–4914 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. La Guardia, G.G.: Quantum Error Correction. Springer Nature, Cham (2020)

    Book  MATH  Google Scholar 

  26. Ling, S., Xing, C.P.: Coding Theory—A First Course. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  27. Lisoněk, P., Singh, V.: Quantum codes from nearly self-orthogonal quaternary linear codes. Des. Codes Cryptogr. 73(2), 417–424 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, X., Hu, P.: New quantum codes from two linear codes. Quantum Inf. Process. 19, 78 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Liu, X., Liu, H.: Quantum codes from linear codes over finite chain rings. Quantum Inf. Process. 16, 240 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

  31. Özbudak, F., Stichtenoth, H.: Note on Niederreiter-Xing’s propagation rule for linear codes. Appl. Algebra Eng. Commun. Comput. 13(1), 53–56 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Prakash, O., Islam, H., Patel, S., Sol\(\acute{e}\), P.: New quantum codes from skew constacyclic codes over a class of non-chain rings \(R_{e,q}\). Int. J. Theor. Phys. 60, 3334–3352 (2021)

  33. Qian, J., Zhang, L.: Improved constructions for quantum maximum distance separable codes. Quantum Inf. Process. 16, 20 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Röttler, M., Grassl, M., Beth, T.: On quantum MDS codes. In: Proceedings of the International Symposium on Information Theory, Chicago, USA, p. 356 (2004)

  35. Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), 2493–2496 (1995)

    Article  ADS  Google Scholar 

  36. Steane, A.M.: Multiple particle interference and quantum error correction. Proc. R. Soc. Lond. A 452, 2551–2557 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Tang, Y., Yao, T., Sun, Z., Zhu, S., Kai, X.: Nonbinary quantum codes from constacyclic codes over polynomial residue rings. Quantum Inf. Process. 19, 84 (2020)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. Verma, R.K., Prakash, O., Singh, A., Islam, H.: New quantum codes from skew constacyclic codes. Adv. Math. Commun. (2021). https://doi.org/10.3934/amc.2021028

    Article  MATH  Google Scholar 

  39. Wang, Y., Kai, X., Sun, Z., Zhu, S.: Quantum codes from Hermitian dual-containing constacyclic codes over \(\mathbb{F} _{q^2}+ v\mathbb{F} _{q^2}\). Quantum Inf. Process. 20, 122 (2021)

    Article  ADS  Google Scholar 

  40. Zhang, X.: Good rate QECCs from the quantum construction X. Quantum Inf. Process. 22, 1 (2023)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by Research Funds of Hubei Province (Grant No. Q20164505) and the talent project of Hubei Polytechnic University of China (Grant No. 16xjzo8R).

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Authors and Affiliations

  1. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, Hubei, China

    Peng Hu

  2. School of Science and Technology, College of Arts and Science of Hubei Normal University, Huangshi, 435109, Hubei, China

    Xiusheng Liu

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  1. Peng Hu
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Correspondence to Xiusheng Liu.

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Hu, P., Liu, X. Quantum error-correcting codes from the quantum construction X. Quantum Inf Process 22, 366 (2023). https://doi.org/10.1007/s11128-023-04122-x

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