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A class of Hermitian dual-containing constacyclic codes and related quantum codes

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Abstract

In this paper, we study a class of Hermitian dual-containing constacyclic codes of length \(n=\frac{q^{2m}-1}{q+1}\), where q is a power of 2 and \(m \ge 2\) is an integer. We apply the Hermitian construction to gain quantum constacyclic codes, and lots of codes obtained in the present paper have parameters better than those of the known quantum codes. The Hermitian hull of a linear code is defined to be the intersection of itself and its Hermitian dual. We construct a family of entanglement-assisted quantum error-correcting (EAQEC) codes with \(n=\frac{4^m-1}{3}\) via determining the dimensions of the Hermitian Hulls.

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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  Google Scholar 

  2. Ashikhim, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)

    Article  Google Scholar 

  3. 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  Google Scholar 

  4. Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474–1484 (2015)

    Article  MathSciNet  Google Scholar 

  5. Du, Z., Li, C., Mesnager, S.: Constuctions of self-orthogonal codes from hulls of BCH codes and their parameters. IEEE Trans. Inf. Theory 66(11), 6774–6785 (2020)

    Article  Google Scholar 

  6. Fan, J., Chen, H., Xu, J.: Constructions of \(q\)-ary entanglement-assisted quantum MDS codes with minimum distance greater than \(q + 1\). Quantum Inf. Comput. 16, 423–434 (2016)

  7. Fang, W., Fu, F.: Two new classes of quantum MDS codes. Finite Fields Appl. 53(6), 85–198 (2018)

    Article  MathSciNet  Google Scholar 

  8. Fang, W., Fu, F., Li, L., Zhu, S.: Euclidean and Hermitian hulls of MDS codes and their applications to EAQECCs. IEEE Trans. Inf. Theory 66(6), 3527–3527 (2020)

    Article  MathSciNet  Google Scholar 

  9. Galindo, C., Hernando, F., Matsumoto, R., Ruano, D.: Entanglement-assisted quantum error-correcting codes over arbitrary finite fields. Quantum Inf. Process. 18(116), 1–18 (2019)

    MathSciNet  Google Scholar 

  10. Gan, C., Li, C., Mesnager, S., Qian, H.: On Hulls of Some Primitive BCH Codes and Self-Orthogonal Codes. IEEE Trans. Inf. Theory 67(10), 6442–6455 (2021)

    Article  MathSciNet  Google Scholar 

  11. Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. Accessed on 2022-11-17. Available from: http://www.codetables.de

  12. Guenda, K., Gulliver, T.A., Jitman, S., Thipworawimon, S.: Linear \(l\)-intersection pairs of codes and their applications. Des. Codes Cryptogr. 88(1), 133–152 (2020)

  13. Guenda, K., Jitman, S., Gulliver, T.A.: Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Cryptogr. 86(1), 121–136 (2018)

    Article  MathSciNet  Google Scholar 

  14. Hu, L., Yue, Q., He, X.: Quantum MDS codes from BCH constacyclic codes. Quantum Inf. Process. 17(12), 1–16 (2018)

    Article  MathSciNet  Google Scholar 

  15. Jin, R., Xie, D., Luo, J.: New classes of entanglement-assisted quantum MDS codes. Quantum Inf. Process. 19(9), 1–12 (2020)

    Article  MathSciNet  Google Scholar 

  16. Kai, X., Zhu, S.: Quantum negacyclic codes. Phys. Rev. A 88, 012326 (2013)

    Article  ADS  Google Scholar 

  17. Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inform. Theory 60(4), 2080–2086 (2014)

    Article  MathSciNet  Google Scholar 

  18. Ketkar, A., Klappenecker, A., Kumar, S.: Nonbinary stablizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)

    Article  Google Scholar 

  19. Key, J.D., Assmus, E.F., Jr.: Affine and projective planes. Discrete Math. 83(2–3), 161–187 (1990)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. La Guardia, G.G.: On optimal constacyclic codes. Linear Algebra Appl. 496, 594–610 (2016)

    Article  MathSciNet  Google Scholar 

  22. Li, C.: Hermitian LCD codes from cyclic codes. Des. Codes Cryptogr. 86(10), 2261–2278 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  23. Li, F., Sun, X.: The Hermitian dual containing non-primitive BCH codes. IEEE Commun. Lett. 25(2), 379–382 (2021)

    Article  ADS  Google Scholar 

  24. Li, F., Yue, Q.: New quantum MDS-convolutional codes derived from constacyclic codes. Mod. Phys. Lett. B 29, 1550252–1550263 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  25. Li, L., Zhu, S., Liu, L.: Three new classes of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes. Quantum Inf. Process. 18(12), 1–16 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  26. Li, R., Li, X.: Binary construction of quantum codes of minimum distance three and four. IEEE Trans. Inf. Theory 50(6), 1331–1336 (2004)

    Article  MathSciNet  Google Scholar 

  27. Li, R., Wang, J., Liu, Y., Guo, G.: New quantum constacyclic codes. Quantum Inf. Process. 18(5), 127 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  28. Li, R., Zuo, F., Liu, Y.: A study of skew symmetric \(q^2\)-cyclotomic coset and its application. J. Air Force Eng. Univ. 12(1), 87–89 (2011)

  29. Li, R., Zuo, F., Liu, Y., Xu, Z.: Hermitian dual-containing BCH codes and construction of new quantum codes. Quantum Inf. Comput. 13(2), 0021–0035 (2013)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Liu, Y., Li, R., Guo, G., Wang, J.: Some Nonprimitive BCH Codes and Related Quantum Codes. IEEE Trans. Inf. Theory 65(12), 7829–7839 (2019)

    Article  MathSciNet  Google Scholar 

  32. Liu, Y., Li, R., Lv, L., Ma, Y.: A class of constacyclic BCH codes and new quantum codes. Quantum Inf. Process. 16(3), 1–16 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  33. Luo, G., Cao, X.: Two new families of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes. Quantum Inf. Process. 18(3), 89 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  34. Luo, G., Cao, X., Chen, X.: MDS codes with hulls of arbitrary dimensions and their quantum error correction. IEEE Trans. Inf. Theory 65(5), 6774–6785 (2019)

    Article  MathSciNet  Google Scholar 

  35. Qian, J., Zhang, L.: Constructions of new entanglement-assisted quantum MDS and almost MDS codes. Quantum Inf. Process. 18(3), 1–12 (2019)

    Article  MathSciNet  Google Scholar 

  36. Shi, X., Yue, Q., Zhu, X.: Construction of some new quantum MDS codes. Finite Fields Appl. 46, 347–362 (2017)

    Article  MathSciNet  Google Scholar 

  37. Song, H., Li, R., Wang, J., Liu, Y.: Two families of BCH codes and new quantum codes. Quantum Inf. Process. 17(10), 1–24 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  38. Tang, N., Li, Z., Xing, L., Zhang, M., Zhao, F.: Some improved constructions for nonbinary quantum BCH codes. IEICE Trans. Fund. E102—-A(1), 303–306 (2019)

    Article  Google Scholar 

  39. Wang, L., Sun, Z., Zhu, S.: Hermitian dual-containing narrow-sense constacyclic BCH codes and quantum codes. Quantum Inf. Process. 18(10), 1–40 (2019)

    Article  MathSciNet  Google Scholar 

  40. Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf. Process. 14(3), 881–889 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  41. Yuan, J., Zhu, S., Kai, X., Li, P.: On the construction of quantum constacyclic codes. Des. Codes Cryptogr. 85(1), 179–190 (2017)

    Article  MathSciNet  Google Scholar 

  42. Zhang, G., Chen, B.: New quantum MDS codes. Int. J. Quantum Inf. Comput. 12(4), 5551–5554 (2014)

    MathSciNet  Google Scholar 

  43. Zhang, M., Li, Z., Xing, L., Tang, N.: Some families of quantum BCH codes. Int. J. Theor. Phys. 58(2), 615–630 (2019)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  45. Zhao, X., Li, X., Wang, Q., Yan, T.: A family of Hermitian dual-containing constacyclic codes and related quantum codes. Quantum Inf. Process. 20, 186 (2021)

  46. Zhu, S., Sun, Z., Li, P.: A class of negacyclic BCH codes and its application to quantum codes. Des. Codes Cryptogr. 86(10), 1–27 (2018)

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Acknowledgements

This study is supported by the National Natural Science Foundation of China (No.61972126, No.U21A20428)

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Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230601, China

    Ping Li, Xiaojing He, Xiaoshan Kai & Jin Li

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  1. Ping Li
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  2. Xiaojing He
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  3. Xiaoshan Kai
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  4. Jin Li
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Correspondence to Xiaojing He.

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Li, P., He, X., Kai, X. et al. A class of Hermitian dual-containing constacyclic codes and related quantum codes. Quantum Inf Process 22, 420 (2023). https://doi.org/10.1007/s11128-023-04171-2

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  • DOI: https://doi.org/10.1007/s11128-023-04171-2

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