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Two Families of Entanglement-Assisted Quantum MDS Codes from Cyclic Codes

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Abstract

With entanglement-assisted (EA) formalism, arbitrary classical linear codes are allowed to transform into EAQECCs by using pre-shared entanglement between the sender and the receiver. In this paper, based on classical cyclic MDS codes by exploiting pre-shared maximally entangled states, we construct two families of q-ary entanglement-assisted quantum MDS codes \( \left[\left[\frac{q^2+1}{a},\frac{q^2+1}{a}-2\left(d-1\right)+c,d;c\right]\right] \), where q is a prime power in the form of am + l, and a = (l2 + 1) or \( a=\frac{\left({l}^2+1\right)}{5} \). We show that all of q-ary EAQMDS have minimum distance upper limit much larger than the known quantum MDS (QMDS) codes of the same length. Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.

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References

  1. Calderbank, A., Shor, P.: Good quantum error-correcting codes exist. Phys rev. Atom mol opt phys. 54, 1098–1105 (1996)

    Article  ADS  Google Scholar 

  2. Shor, P.: Scheme for reducing decoherence in quantum computer memory. Phys rev. Atomic mol opt phys. 52, 2493–2496 (1995)

    Article  ADS  Google Scholar 

  3. Hsieh, M., Brun, T., Devetak, I.: Correcting quantum errors with entanglement. Sci. 314, 436–439 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  4. Brun, T., Lai, C., Wilde, M.: Duality in entanglement-assisted quantum error correction. IEEE Trans. Inf. Theory. 59(6), 4020–4024 (2013)

    Article  MathSciNet  Google Scholar 

  5. Brun, T., Lai, C., Wilde, M.: Dualities and identities for entanglement-assisted quantum codes. Quantum Inf. Process. 13, 957–990 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  6. Fujiwara, Y., Clark, D., Vandendriessche, P., Boeck, M., Tonchev, V.: Entanglement-assisted quantum low-density parity-check codes. Phys. Rev. A. 82(4), 272–277 (2010)

    Article  Google Scholar 

  7. Guo, L., Li, R.: Linear plotkin bound for entanglement-assisted quantum codes. Phys. Rev. A. 87(3), 1764–1770 (2013)

    Article  Google Scholar 

  8. Hsieh, M., Yen, W., Hsu, L.: High performance entanglement-assisted quantum ldpc codes need little entanglement. IEEE Trans. Inf. Theory. 57(3), 1761–1769 (2011)

    Article  MathSciNet  Google Scholar 

  9. Koroglu, M.: New entanglement-assisted mds quantum codes from constacyclic codes. Quantum Inf. Process. 18, 1–28 (2019)

    Article  MathSciNet  Google Scholar 

  10. Lai, C., Brun, T.: Entanglement increases the error-correcting ability of quantum error-correcting codes. Phys. Rev. A. 88(1), 2343–2347 (2010)

    Google Scholar 

  11. Lu, L., Li, R.: Entanglement-assisted quantum codes constructed from primitive quaternary bch codes. Int J Quantum Inf. 12(03), 1450015 (2014)

    Article  MathSciNet  Google Scholar 

  12. Brun, T., Hsieh, M., Devetak, I.: General entanglement-assisted quantum error-correcting codes. Phys. Rev. A. 76, 062313 (2007)

    Article  ADS  Google Scholar 

  13. Hsieh, M., Wilde, M., Babar, Z.: Entanglement-assisted quantum turbo codes. 2014

  14. Brun, T., Wilde, M.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A. 77, 064302 (2008)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. Kan, H., Jin, L., Wen, J.: Quantum Mds Codes with Relatively Large Minimum Distance from Hermitian Self-Orthogonal Codes. Designs, Codes and Cryptography, 2017

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

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  20. Matsumoto, R., Galindo, C., Hernando, F., Diego, R.: Entanglement-assisted quantum error-correcting codes over arbitrary finite fields. Quantum Information Processing, 18(4), 2019

  21. Zhu, S., Tian, F.: Some new entanglement-assisted quantum error-correcting mds codes from generalized reed–solomon codes. Quantum Information Processing, 19(7), 2020

  22. Feng, C., Chen, J., Huang, Y., Chen, R.: Entanglement-assisted quantum mds codes constructed from negacyclic codes. Quantum Inf. Process. 16(12), 1–22 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  23. Gulliver, T., Guenda, K., Jitman, S.: Constructions of good entanglement-assisted quantum error correcting codes. Designs, Codes and Cryptography. 86, 121–136

  24. Luo, G., Cao, X., Chen, X.: Mds codes with hulls of arbitrary dimensions and their quantum error correction. IEEE Trans. Inf. Theory. 65, 2944–2952 (2019)

    Article  MathSciNet  Google Scholar 

  25. Pang, B., Zhu, S., Li, F., Chen, X.: New entanglement-assisted quantum mds codes with larger minimum distance. Quantum Information Processing, 19(7), 2020

  26. Qian, J., Zhang, L.: Entanglement-assisted quantum codes from arbitrary binary linear codes. Designs Codes & Crytography. 77(1), 193–202 (2015)

    Article  MathSciNet  Google Scholar 

  27. Qian, J., Zhang, L.: On mds linear complementary dual codes and entanglement-assisted quantum codes. Designs Codes & Cryptography. 86(7), 1565–1572 (2018)

    Article  MathSciNet  Google Scholar 

  28. Qian, J., Zhang, L.: Constructions of new entanglement-assisted quantum mds and almost mds codes. Quantum Information Processing, 18(3), 2019

  29. Lu, Y., Liu, Li, R., Ma, Y.: Application of constacyclic codes to entanglement-assisted quantum maximum distance separable codes. Quantum Inf Process. 17(8), 210 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  30. Zhu, S., Jiang, W., Chen, X.: New entanglement-assisted quantum mds codes with length \( n=\frac{q^2+1}{5} \). Quantum Inf Process, 19(7):1–15, 2020

  31. Lu, L., Ma, W., Li, R., Ma, Y., Liu, Y., Cao, H.: Entanglement-assisted quantum mds codes from constacyclic codes with large minimum distance. Finite Fields Thr Appl. 53, 309–325 (2018)

    Article  MathSciNet  Google Scholar 

  32. Lu, L., Guo, L., Li, R., Ma, Y., Liu, Y.: Entanglement-assisted quantum mds codes from negacyclic codes. Quantum Inf Process. 17(3), 69 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  33. Zhu, S. Chen, X., Kai, X.: Entanglement-assisted quantum mds codes constructed from constacyclic codes. Quantum Information Processing, 17(10), 2018

  34. Liu, Y., Li, R., Zuo, F.: A study of skew symmetric q 2-cyclotomic coset and its application. Journal of Air Force Engineering University (Natural ence Edition), 2011

  35. Huffman, W., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, 2003

  36. Macwilliams, F. J., Alexander, N.J. Sloane, N.J.A.: The Theory of Error-Correcting Codes. N.H.P.C, 1977

  37. Liu, Y., Li, R., Zuo, F., Xu, Z.: Hermitian dual containing bch codes and construction of new quantum codes. Quantum Inf Comput. 13(1–2), 21–35 (2013)

    MathSciNet  Google Scholar 

Download references

Acknowledgments

We are indebted to the anonymous referees for their valuable comments and suggestions that improved the quality of this paper. This work is supported by the National Natural Science Foundation of China under Grant No.11801564, 61373171 and 11901579, the National Key R&D Program of China under Grant No. 2017YFB0802400, 111 Project under grant No.B08038, Shannxi Natural Science Foundation under Grant No. 2021JM-216, 2021JM-335.

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Correspondence to Liangdong Lu.

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Lu, L., Ma, W., Li, R. et al. Two Families of Entanglement-Assisted Quantum MDS Codes from Cyclic Codes. Int J Theor Phys 60, 1833–1842 (2021). https://doi.org/10.1007/s10773-021-04802-3

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  • DOI: https://doi.org/10.1007/s10773-021-04802-3

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