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Asymmetric quantum convolutional codes

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Abstract

In this paper, we construct the first families of asymmetric quantum convolutional codes (AQCCs). These new AQCCs are constructed by means of the CSS-type construction applied to suitable families of classical convolutional codes, which are also constructed here. The new codes have non-catastrophic generator matrices, and they have great asymmetry. Since our constructions are performed algebraically, i.e. we develop general algebraic methods and properties to perform the constructions, it is possible to derive several families of such codes and not only codes with specific parameters. Additionally, several different types of such codes are obtained.

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References

  1. Aly, S.A., Grassl, M., Klappenecker, A., Rötteler, M., Sarvepalli, P.K.: Quantum convolutional BCH codes. arXiv:quant-ph/0703113

  2. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Quantum convolutional codes derived from Reed–Solomon and Reed–Muller codes. arXiv:quant-ph/0701037

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over \(GF(4)\). IEEE Trans. Inform. Theory 44(4), 1369–1387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Forney Jr, G.D.: Convolutional codes I: algebraic structure. IEEE Trans. Inform. Theory 16(6), 720–738 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  6. de Almeida, A.C.A., Palazzo Jr., R.: A concatenated \([(4, 1, 3)]\) quantum convolutional code. In: Proceedings of IEEE Information Theory Workshop (ITW), pp. 28–33 (2004)

  7. Forney Jr, G.D., Grassl, M., Guha, S.: Convolutional and tail-biting quantum error-correcting codes. IEEE Trans. Inform. Theory 53(3), 865–880 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Grassl, M., Rötteler, M.: Non-catastrophic encoders and encoder inverses for quantum convolutional codes. In: Proceedings of International Symposium on Information Theory (ISIT), pp. 1109–1113 (2006)

  9. Grassl, M., Rötteler, M.: Constructions of quantum convolutional codes. arXiv:quant-ph/0703182

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  12. Johannesson, R., Zigangirov, K.S.: Fundamentals of Convolutional Coding. Digital and Mobile Communication. Wiley, New York (1999)

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. La Guardia, G.G.: Asymmetric quantum Reed–Solomon and generalized Reed–Solomon codes. Quantum Inform. Process. 11(2), 591–604 (2012)

    Article  MATH  Google Scholar 

  17. La Guardia, G.G.: Asymmetric quantum codes: new codes from old. Quantum Inform. Process. 12, 2771–2790 (2013)

    Article  ADS  MATH  Google Scholar 

  18. La Guardia, G.G.: On classical and quantum MDS-convolutional BCH codes. IEEE Trans. Inform. Theory 60(1), 304–312 (2014)

    Article  MathSciNet  Google Scholar 

  19. La Guardia, G.G.: On negacyclic MDS-convolutional codes. Linear Algebra Appl. 448, 85–96 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lee, L.N.: Short unit-memory byte-oriented binary convolutional codes having maximum free distance. IEEE Trans. Inform. Theory 22, 349–352 (1976)

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  22. Ollivier, H., Tillich, J.-P.: Description of a quantum convolutional code. Phys. Rev. Lett. 91(17), 1779021–4 (2003)

    Article  Google Scholar 

  23. Ollivier, H., Tillich, J.-P.: Quantum convolutional codes: fundamentals. arXiv:quant-ph/0401134

  24. Piret, Ph: Convolutional Codes: An Algebraic Approach. The MIT Press, Cambridge, Massachusetts (1988)

    MATH  Google Scholar 

  25. Rosenthal, J., Smarandache, R.: Maximum distance separable convolutional codes. Appl. Algebra Eng. Commun. Comput. 10, 15–32 (1998)

    Article  MathSciNet  Google Scholar 

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

  27. Smarandache, R., Luerssen, H.G., Rosenthal, J.: Constructions of MDS-convolutional codes. IEEE Trans. Inform. Theory 47(5), 2045–2049 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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Acknowledgments

This research has been partially supported by the Brazilian Agencies CAPES and CNPq. This article is dedicated to my beloved younger brother Paolo Gadioli La Guardia (La Guardia, P.G.), researcher in medicine, who died in 2014. His soul lives forever with eternal felicities.

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

    Giuliano G. La Guardia

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  1. Giuliano G. La Guardia
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Correspondence to Giuliano G. La Guardia.

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La Guardia, G.G. Asymmetric quantum convolutional codes. Quantum Inf Process 15, 167–183 (2016). https://doi.org/10.1007/s11128-015-1192-9

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