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On quantum SPC product codes

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Abstract

Some methods have been proposed to construct quantum codes out of classical codes. There is a relationship between quantum codes and codes on the quaternary field. The structure of CSS codes has been raised to construct quantum codes. Most of them are based on the fact that a classical code is self-orthogonal. In this paper, a quantum code is constructed using a suitable permutation-based technique on the parity check matrix of classical codes. Several examples of the new quantum error correcting codes are provided. Single parity check (SPC) product codes have simple decoding algorithms, as well as good decoding efficiency. With this technique, the SPC product codes are used to construct quantum codes.

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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  Google Scholar 

  2. Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098–1105 (1996)

    Article  ADS  Google Scholar 

  3. Steane, A.M.: Multiple particle interference and quantum error correction. Proc. R. Soc. Lond. A 452, 2551–2577 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  4. MacKay, D.J.C., Mitchison, G., McFadden, P.L.: Sparse-graph codes for quantum error correcting. IEEE Trans. Inf. Theory 50, 2315–2330 (2004)

    Article  Google Scholar 

  5. Djordjevic, I.B.: Quantum LDPC codes from balanced incomplete block designs. IEEE Commun. Lett. 12, 389–391 (2008)

    Article  Google Scholar 

  6. Aly S.A.: A Class of quantum LDPC codes constructed from finite geometries. In: IEEE GLOBECOM, pp. 1–5 (2008)

  7. Couvreur, A., Delfosse, N., Zmor, G.: A construction of quantum LDPC codes from Cayley graphs. IEEE Trans. Inf. Theory 59, 6087–6098 (2013)

    Article  MathSciNet  Google Scholar 

  8. Munuera, C., Tenorio, W., Torres, F.: Quantum error-correcting codes from algebraic geometry codes of Castle type. Quantum Inf. Process. 15(10), 4071–4088 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  9. Tang, Y., Zhu, S., Kai, X., Ding, J.: New quantum codes from dual-containing cyclic codes over finite rings. Quantum Inf. Process. 15, 4489–4500 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  10. 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 

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

    Article  MathSciNet  Google Scholar 

  12. La Guardia, G.G., Palazzo, R.: Constructions of new families of nonbinary CSS code. Discrete Math. 310(21), 2935–2945 (2010)

    Article  MathSciNet  Google Scholar 

  13. 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 

  14. Hagiwara, M., Imai, H.: Quantum quasi-cyclic LDPC codes. In: Proceedings of IEEE International Symposium on Information Theory, ISIT (2007)

  15. Dennis, E., Kitaev, A., Landahl, A., Preskill, J.: Topological quantum memory. J. Math. Phys. 43, 4452–4505 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  16. Kitaev, A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  17. Bombin, H., Martin-Delgado, M.A.: Homological error correction: classical and quantum codes. J. Math. Phys. 48, 052105 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  18. Elias, P.: Error-free coding. IRE Trans. Inf. Theory 4, 29–37 (1954)

    Article  MathSciNet  Google Scholar 

  19. Grassl, M., Rotteler, M.: Quantum block and convolutional codes from self-orthogonal product codes. In: Proceedings of International Symposium on Information Theory (ISIT) (2005)

  20. Fan, J., Li, Y., Hsieh, M.H., Chen, H.: On quantum tensor product codes. Quantum Inf. Comput. 17, 1105–1122 (2017)

    MathSciNet  Google Scholar 

  21. Guardia, G.G.: Asymmetric quantum product codes. Int. J. Quantum Inf. 10, 1250005 (2012)

    Article  MathSciNet  Google Scholar 

  22. Caire, G., Taricco, G., Battail, G.: Weight distribution and performance of the iterated product of single-parity-check codes. In: IEEE GLOBECOM, pp. 206–211 (1994)

  23. Biglieri, E., Volski, V.: Approximately Gaussian weight distribution of the iterated product of single-parity-check codes. Electron. Lett. 30(12), 923–924 (1994)

    Article  Google Scholar 

  24. Ping, L., Chan, S., Yeung, K.: Efficient soft-in-soft-out sub-optimal decoding rule for single parity check codes. Electron. Lett. 33(19), 1614–1616 (1997)

    Article  Google Scholar 

  25. Ping, L., Chan, S., Yeung, K.L.: Iterative decoding of multi-dimensional concatenated single parity check codes. In: IEEE International Conference on Communications, pp. 131–135 (1998)

  26. Huang, X., Phamdo, N., Ping, L.: BER bounds on parallel concatenated single parity check arrays and zigzag codes. In: Conference GLOBECOM, pp. 2436–2440 (1999)

  27. Rankin, D.M., Gulliver, T.A.: Single parity check product codes. IEEE Trans. Commun. 49(8), 1354–1362 (2001)

    Article  Google Scholar 

  28. Kousa, M.A.: A novel approach for evaluating the performance of SPC product codes under erasure decoding. IEEE Trans. Commun. 50(1), 7–11 (2002)

    Article  Google Scholar 

  29. Amutha, R., Verraraghavan, K., Srivatsa, S.K.: Recoverability study of SPC product codes under erasure decoding. Inf. Sci. 173, 169–179 (2005)

    Article  MathSciNet  Google Scholar 

  30. Coskun, M.C., Liva, G., Graell, A., Amat, I., Lentmaier, M.: Successive cancellation decoding of single parity-check product codes. In: IEEE International Symposium on Information Theory (ISIT) (2017)

  31. Gottesman, D.: Stabilizer codes and quantum error correction, Ph.D. thesis, Caltech (1997)

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

    MATH  Google Scholar 

  33. Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 54, R2493 (1995)

    Article  ADS  Google Scholar 

  34. Tonchev, V.D.: Error-correcting codes from graphs. Discrete Math. 257, 549–557 (2002)

    Article  MathSciNet  Google Scholar 

  35. Roth, R.M.: Introduction to Coding Theory. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  36. Hivadi, M., Esmaeili, M.: On the stopping distance and stopping redundancy of product codes. IEICE Trans. E91–A, 2167–2173 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

I am indebted to the anonymous referees for their comments and suggestions that have improved the quality of this paper.

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  1. Department of Mathematical Sciences, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran

    M. Hivadi

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  1. M. Hivadi
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Hivadi, M. On quantum SPC product codes. Quantum Inf Process 17, 324 (2018). https://doi.org/10.1007/s11128-018-2095-3

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