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Entanglement-assisted quantum error correction codes with length \(n=q^2+1\)

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Abstract

In this paper, by investigating \(q^2\)-cyclotomic coset modulo rn in detail, where q is a prime power, \(n=q^2+1\) and \(r\mid (q+1)\), series of entanglement-assisted quantum error correction (EAQEC) codes with flexible parameters of length n are constructed from constacyclic codes (including cyclic codes). Most of our EAQEC codes are new and have large minimum distance. As to EAQEC codes constructed from cyclic codes, their all possible parameters are determined completely. When minimum distance \(d\le \frac{n+2}{2}\), all of our constructed EAQEC codes are entanglement-assisted quantum MDS (EAQMDS) codes. Those previously known EAQMDS codes with the same length in Fan et al. (Quantum Inf Comput 16:423–434, 2016), Chen et al. (Quantum Inf Process 16(303):1–22, 2017), Lu et al. (Finite Fields Their Appl 53:309–325, 2018), Mustafa and Emre (Comput Appl Math 38(75):1–13, 2019) and Qian and Zhang (Quantum Inf Process 18(71):1–12, 2019) are special cases of ours. Besides, some maximum entanglement EAQEC codes and maximum entanglement EAQMDS codes are derived as well.

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References

  1. 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, 1369–1387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Grassl, M., Beth, T.: Quantum BCH codes. In: Proceedings X. International Symposium on Theoretical Electrical Engineering Magdeburg, pp. 207–212 (1999)

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, R., Zuo, F., Liu, Y., Xu, Z.: Hermitian dual-containing BCH codes and construction of new quantum codes. Quantum Inf. Comput. 12, 0021–0035 (2013)

    MathSciNet  Google Scholar 

  8. Liu, Y., Li, R., Lv, L., Ma, Y.: A class of constacyclic BCH codes and new quantum codes. Quantum Inf. Process. 16(66), 1–16 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  9. Song, H., Li, R., Wang, J., Liu, Y.: Two classes of BCH codes and new quantum codes. Quantum Inf. Process. 17(270), 1–24 (2018)

    ADS  MATH  Google Scholar 

  10. Li, R., Wang, J., Liu, Y., Guo, G.: New quantum constacyclic codes. Quantum Inf. Process. 18(127), 1–23 (2019)

    ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Jin, L., Xing, C.: A construction of new quantum MDS codes. IEEE. Trans. Inf. Theory 60, 2921–2925 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grassl, M., Rtteler, M.: Quantum MDS codes over small fields. In: IEEE IEEE International Symposium on Information Theory, pp. 1104–1108(2015)

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Grassl, M.: Entanglement-assisted quantum communication beating the quantum singleton bound. In: AQIS, Taiwan (2016)

  18. 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)

    MathSciNet  Google Scholar 

  19. Chen, J., Huang, Y., Feng, C., Chen, R.: Entanglement-assisted quantum MDS codes constructed from negacyclic codes. Quantum Inf. Process. 16(303), 1–22 (2017)

    ADS  MathSciNet  MATH  Google Scholar 

  20. 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 Their Appl. 53, 309–325 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mustafa, S., Emre, K.: An application of constacyclic codes to entanglement-assisted quantum MDS codes. Comput. Appl. Math. 38(75), 1–13 (2019)

    MathSciNet  Google Scholar 

  22. Qian, J., Zhang, L.: On MDS linear complementary dual codes and entanglement-assisted quantum codes. Des. Codes Cryptogr. 86, 1565–1572 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Qian, J., Zhang, L.: Constructions of new entanglement-assisted quantum MDS codes and almost MDS codes. Quantum Inf. Process. 18(71), 1–12 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Li, R., Guo, G., Song, H., Liu, Y.: New constructions of entanglement-assisted quantum MDS codes from negacyclic codes. Int. J. Quantum Inf. 17(1), 1950022 (2019)

    Article  MathSciNet  Google Scholar 

  25. Liu, Y., Li, R., Lv, L., Ma, Y.: Application of constacyclic codes to entanglement-assisted quantum maximum diatance separable codes. Quantum Inf. Process. 17(210), 1–19 (2018)

    Article  ADS  MATH  Google Scholar 

  26. Fang, W., Fu, F., Li, L., Zhu, S.: Euclidean and Hermitian hulls of MDS codes and their applications to EAQECCs (2018)

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

    Article  ADS  Google Scholar 

  28. Macwilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North-Holland Publishing Company, Amsterdam (1977)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  30. Aydin, N., Siap, I., Ray-Chaudhuri, D.K.: The structure of 1-generator quasi-twisted codes and new linear codes. Des. Codes Cryptogr. 24, 313–326 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Krishna, A., Sarwate, D.V.: Pseudo-cyclic maximum-distance separable codes. IEEE. Trans. Inf. Theory 36, 880–884 (1990)

    Article  MATH  Google Scholar 

  32. Lü, L., Li, R.: Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes. Int. J. Quantum Inf. 12(3), 1450015 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Basic Sciences, Air Force Engineering University, Xi’an, 710051, Shaanxi, People’s Republic of China

    Junli Wang, Ruihu Li, Jingjie Lv, Guanmin Guo & Yang Liu

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  1. Junli Wang
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  2. Ruihu Li
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  3. Jingjie Lv
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  4. Guanmin Guo
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  5. Yang Liu
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Correspondence to Ruihu Li.

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This work is supported by National Natural Science Foundation of China under Grant Nos. 11471011, 11801564 and Natural Science Foundation of Shaanxi under Grant No. 2017JQ1032.

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Wang, J., Li, R., Lv, J. et al. Entanglement-assisted quantum error correction codes with length \(n=q^2+1\). Quantum Inf Process 18, 292 (2019). https://doi.org/10.1007/s11128-019-2409-0

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