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Some Families of Quantum BCH Codes

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Abstract

Classical Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields have been studied extensively. The Calderbank-Shor-Steane (CSS) construction, especially Steane’s enlargement, and Hermitian construction are the most widely used methods in design of quantum codes. The BCH codes containing their Euclidean dual or Hermitian dual codes can be used to generate good stabilizer codes. Therefore, we can construct quantum codes by classical BCH codes over finite fields in this paper. Firstly, we study the properties of such classical BCH codes in terms of the cyclotomic cosets. It is convenient to compute the dimension of new quantum BCH codes. Meanwhile, it ensures that classical BCH codes are Euclidean dual-containing or Hermitian dual-containing. These results about suitable cyclotomic cosets make it possible to construct several new families of nonbinary quantum BCH codes with a given parameter set. Compared with the ones available in the literature, the quantum BCH codes in our schemes have good parameters. In particular, we extend to more general cases than known results.

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References

  1. La Guardia, G.G.: On the construction of nonbinary quantum BCH codes. IEEE Trans. Inf. Theory 60(3), 1528–1535 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Hu, X.Q., Zhang, G.H., Chen, B.C.: Constructions of new nonbinary quantum codes. Int. J. Theor. Phys. 54(1), 92–99 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Gao, J., Wang, Y.K.: Quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 57(3), 682–686 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. The Netherlands (1977)

  5. Charpin, P.: Open Problems on Cyclic Codes in Handbook of Coding Theory, pp. 963–1063. Amsterdam (1998)

  6. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge Univ Press, Cambridge (2000)

    MATH  Google Scholar 

  7. 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 

  8. Ling, S., Luo, J.Q., Xing, C.P.: Generalization of Steane’s enlargement construction of quantum codes and applications. IEEE Trans. Inf. Theory 56(8), 4080–4084 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. La Guardia, G.G.: Quantum codes derived from cyclic codes. Int. J. Theor. Phys. 56(8), 2479–2484 (2017)

    Article  MATH  Google Scholar 

  10. Ma, Z., Lu, X., Feng, K.Q., Feng, D.G.: On non-binary quantum BCH codes. LNCS 3959, 675–683 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Ma, Y.N., Xu, G.: Two families of BCH codes and construction of quantum codes. In: 2014 International Conference on Information Science, Electronics and Electrical Engineering, pp. 249–252. Sapporo (2014)

  12. La Guardia, G.G.: New quantum MDS codes. IEEE Trans. Inf. Theory 57(8), 5551–5554 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. IEEE Int. Symp. Inf. Theory, 1105–1108 (2006)

  14. La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(4), 042331(1–11) (2009)

    Article  ADS  Google Scholar 

  15. Kai, X.S., Zhu, S.X.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 5551–5554 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grassl, M., Rötteler, M.: Quantum MDS codes over small fields. IEEE Int. Symp. Inf. Theory, 1104–1108 (2015)

  17. Zhang, T., Ge, G.N.: Some new classes of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61(9), 5224–5228 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Zhang, G.H., Chen, B.C., Li, L.C.: A construction of MDS quantum convolution codes. Int. J. Theor. Phys. 54(9), 3182–3194 (2015)

    Article  MATH  Google Scholar 

  19. Huang, Y.Y., Chen, J.J.Z., Feng, C.H., Chen, R.Q.: Some families of asymmetric quantum MDS codes constructed from constacyclic codes. Int. J. Theor. Phys. 57(2), 453–464 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Qian, J.F., Zhang, L.N.: Improved constructions for nonbinary quantum BCH codes. Int. J. Theor. Phys. 56(4), 1355–1363 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, J.Z., Li, J.P., Lin, J.: New optimal asymmetric quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 53(1), 72–79 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, M., Li, Z, Xing, L.J., Tang, N.Q.: Construction of some new quantum BCH codes. IEEE Access 6(1), 36122–36131 (2018)

    Article  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61372072, in part by the 111 Project under Grant B08038, and in part by the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an, Shannxi, 710071, China

    Ming Zhang, Zhuo Li, Lijuan Xing & Nianqi Tang

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  1. Ming Zhang
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  2. Zhuo Li
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  3. Lijuan Xing
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  4. Nianqi Tang
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Correspondence to Zhuo Li.

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Zhang, M., Li, Z., Xing, L. et al. Some Families of Quantum BCH Codes. Int J Theor Phys 58, 615–630 (2019). https://doi.org/10.1007/s10773-018-3959-0

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  • DOI: https://doi.org/10.1007/s10773-018-3959-0

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