Skip to main content
SpringerLink
Log in

Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

Two new families of asymmetric quantum codes are constructed in this paper. The first one is derived from the Calderbank-Shor-Steane (CSS) construction applied to classical Reed-Solomon (RS) codes, providing quantum codes with parameters [[N = l(q l−1), K = l(q l−2d + c + 1), d z ≥ d/d x ≥ (d−c)]] q , where q is a prime power and d > c + 1, c ≥ 1, l ≥ 1 are integers. The second family is derived from the CSS construction applied to classical generalized RS codes, generating quantum codes with parameters [[N = mn, K = m(2k−n + c), d z ≥ d/d x ≥ (d−c)]] q , where q is a prime power, 1 < k < n < 2k + c ≤ q m, k = n − d + 1, and n, d > c + 1, c ≥ 1, m ≥ 1 are integers. Although the second proposed construction generalizes the first one, the techniques developed in both constructions are slightly different. These new codes have parameters better than or comparable to the ones available in the literature. Additionally, the proposed codes can be utilized in quantum channels having great asymmetry, that is, quantum channels in which the probability of occurrence of phase-shift errors is large when compared to the probability of occurrence of qudit-flip errors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aly, S.A.: Asymmetric quantum BCH codes. In: Proceedings of IEEE International Conference on Computer Engineering and Systems (ICCES08), pp. 157–162 (2008)

  2. Aly, S.A., Ashikhmin, A.: Nonbinary quantum cyclic and subsystem codes over asymmetrically- decohered quantum channels. e-print arXiv:quant-ph/1002.2966

  3. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. In: Proceedings of International Symposium on Information Theory (ISIT), pp. 1114–1118 (2006)

  4. 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 

  5. Ashikhmin, A., Litsyn, S., Tsfasman, M.A.: Asymptotically good quantum codes. e-print arXiv:quant-ph/0006061

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Bierbrauer J., Edel Y.: Quantum twisted codes. J. Comb. Des. 8, 174–188 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Calderbank A.R., Shor P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1106 (1996)

    Article  ADS  Google Scholar 

  10. Chen H.: Some good error-correcting codes from algebraic-geometric codes. IEEE Trans. Inf. Theory 47(5), 2059–2061 (2001)

    Article  MATH  Google Scholar 

  11. Chen H., Ling S., Xing C.: Quantum codes from concatenated algebraic-geometric codes. IEEE Trans. Inf. Theory 51(8), 2915–2920 (2005)

    Article  MathSciNet  Google Scholar 

  12. Evans, Z.W.E., Stephens, A.M., Cole, J.H., Hollenberg, L.C.L.: Error correction optimisation in the presence of x/z asymmetry. e-print arXiv:quant-ph/0709.3875

  13. Ezerman, M.F., Jitman, S., Ling, S.: On asymmetric quantum MDS codes. e-print arXiv:quant-ph/1006.1694

  14. Ezerman, M.F., Ling, S., Solé, P.: Additive asymmetric quantum codes. e-print arXiv:quant-ph/1002.4088

  15. Ezerman, M.F., Solé, P., Ling, S.: Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. e-print arXiv:quant-ph/1005.0879

  16. Fenn S.T.J., Benaissa M., Taylor D.: GF(2m) multiplication and division over the dual basis. IEEE Trans. Comp. 45(3), 319–327 (1996)

    Article  MATH  Google Scholar 

  17. Grassl M., Beth T., Rötteler M.: On optimal quantum codes. Int. J. Quantum. Inf. 2(1), 757–766 (2004)

    Article  Google Scholar 

  18. Grassl M., Geiselmann W., Beth T.: Quantum Reed-Solomon codes. AAECC-13 1709, 231–244 (1999)

    MathSciNet  Google Scholar 

  19. Hamada M.: Lower bounds on the quantum capacity and highest error exponent of general memoryless channels. IEEE Trans. Inf. Theory 48(9), 2547–2557 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hamada, M.: Quotient codes and their reliability. IPSJ Digit. Cour. 1, 450–460 (2005)

  21. Hamada, M.: Conjugate codes and applications to cryptography. e-print arXiv:quant-ph/0610193

  22. Hamada, M.: Conjugate codes for secure and reliable information transmission. In: Proceedings of 2006 IEEE Information Workshop (ITW’06), pp. 149–153 (2006)

  23. Hamada M.: Concatenated quantum codes constructible in polynomial time: efficient decoding and error correction. IEEE Trans. Inf. Theory 54(12), 5689–5704 (2008)

    Article  MathSciNet  Google Scholar 

  24. Ioffe L., Mezard M.: Asymmetric quantum error-correcting codes. Phys. Rev. A 75, 032345-1–032345-4 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  25. 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  Google Scholar 

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

    Article  ADS  Google Scholar 

  27. La Guardia G.G., Palazzo R. Jr: Constructions of new families of nonbinary CSS codes. Discret. Math. 310(21), 2935–2945 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. La Guardia G.G., Palazzo R. Jr, Lavor C.: Nonbinary quantum Reed-Solomon codes. Int. J. Pure Appl. Math. 65(1), 55–63 (2010)

    MathSciNet  MATH  Google Scholar 

  29. La Guardia G.G.: New families of asymmetric quantum BCH codes. Quantum Inf. Comput. 11(3-4), 239–252 (2011)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  31. Li Z., Xing L.J., Wang X.M.: Quantum generalized Reed-Solomon codes: unified framework for quantum maximum-distance-separable codes. Phys. Rev. A 77, 012308-1–012308-4 (2008)

    MathSciNet  ADS  Google Scholar 

  32. Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997)

    Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  35. Sarvepalli, P.K., Klappenecker, A., Rötteler, M.: Asymmetric quantum LDPC codes. In: Proceedings of International Symposium on Information Theory (ISIT), pp. 6–11 (2008)

  36. Sarvepalli, P.K., Klappenecker, A., Rötteler, M.: Asymmetric quantum codes: constructions, bounds and performance. In: Proceedings of of the Royal Society A, pp. 1645–1672 (2009)

  37. Steane A.M.: Simple quantum error correcting-codes. Phys. Rev. A 54, 4741–4751 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  38. Steane A.M.: Enlargement of Calderbank-Shor-Steane quantum codes. IEEE Trans. Inf. Theory 45(7), 2492–2495 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Stephens A.M., Evans Z.W.E., Devitt S.J., Hollenberg L.C.L.: Asymmetric quantum error correction via code conversion. Phys. Rev. A 77, 062335-1–062335-5 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  40. Wang L., Feng K., Ling S., Xing C.: Asymmetric quantum codes: characterization and constructions. IEEE Trans. Inf. Theory 56(6), 2938–2945 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, State University of Ponta Grossa, Ponta Grossa, PR, 84030-900, Brazil

    Giuliano G. La Guardia

Authors
  1. Giuliano G. La Guardia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giuliano G. La Guardia.

Rights and permissions

Reprints and permissions

About this article

Cite this article

La Guardia, G.G. Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes. Quantum Inf Process 11, 591–604 (2012). https://doi.org/10.1007/s11128-011-0269-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-011-0269-3

Keywords

Search

Navigation