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New optimal asymmetric quantum codes and quantum convolutional codes derived from constacyclic codes

  • Jianzhang Chen
  • Youqin Chen
  • Yuanyuan HuangEmail author
  • Chunhui Feng
Article
  • 36 Downloads

Abstract

In this paper, some families of asymmetric quantum codes and quantum convolutional codes that satisfy the quantum Singleton bound are constructed by utilizing constacyclic codes with length \(n=\frac{q^2+1}{10h}\), where q is an odd prime power with the form \(q=10hm+t\) or \(q=10hm+10h-t\), where m is a positive integer, and both h and t are odd with \(10h=t^2+1\) and \(t\ge 3\). Compared with those codes constructed in the literature, the parameters of these constructed quantum codes in this paper are more general. Moreover, the distance \(d_z\) of optimal asymmetric quantum codes \([[n,k,d_z/d_x]]_{q^2}\) here is larger than most of the ones given in the literature.

Keywords

Constacyclic codes Asymmetric quantum codes Quantum convolutional codes Quantum Singleton bound 

Notes

Acknowledgements

The research was supported by the Natural Science Foundation of China (No. 61802064) and the Natural Science Foundation of Fujian Province, China (Nos. 2016J01281, 2016J01278). We are indebted to anonymous reviewers who have made constructive suggestions for the improvement of this manuscript.

References

  1. 1.
    Aly, S.A.: Asymmetric quantum BCH codes. In: Proceedings International Conference on Computer Engineering System, pp. 157–162 (2008)Google Scholar
  2. 2.
    Aly, S.A., Grassl, M., Klappenecker, A., Rötteler, M., Sarvepalli, P.K.: Quantum convolutional BCH codes. In: Proceedings of 10th Canadian Workshop on Information Theory, pp. 180–183 (2007)Google Scholar
  3. 3.
    Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Quantum convolutional codes derived from Reed–Solomon and Reed–Muller codes. arXiv:quant-ph/0701037
  4. 4.
    Ashikhmin, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)CrossRefGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bakshi, G.K., Raka, M.: A class of constacyclic codes over a finite field. Finite Fields Appl. 18, 362–377 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Berlekamp, E.R.: Negacyclic codes for the Lee metric. In: Proceedings of Symposium in Combinatorial Mathematics and Its Applications, pp. 1–27 (1967)Google Scholar
  8. 8.
    Blackford, T.: Negacyclic duadic codes. Finite Fields Appl. 14, 930–943 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, B., Fan, Y., Lin, L., Liu, H.: Constacyclic codes over finite fields. Finite Fields Appl. 18, 1217–1231 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474–1484 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, J., Li, J., Lin, J.: New optimal asymmetric quantum codes derived from negacyclic codes. Int. J. Theor. Phys. 53(1), 72–79 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, J., Huang, Y., Feng, C., Chen, R.: Some families of optimal quantum codes derived from constacyclic codes. Linear Multilinear Algebra (2018).  https://doi.org/10.1080/03081087.2018.1432544
  14. 14.
    Chen, J., Li, J., Yang, F., Huang, Y.: Nonbinary quantum convolutional codes derived from negacyclic nodes. Int. J. Theor. Phys. 54(1), 198–209 (2015)CrossRefGoogle Scholar
  15. 15.
    Chen, J., Lin, J., Huang, Y.: Asymmetric quantum codes and quantum convolutional codes derived from nonprimitive non-narrow-sense BCH codes. IEICE Trans. Fund. Electr. 98(5), 1130–1135 (2015)CrossRefGoogle Scholar
  16. 16.
    Chen, J., Li, J., Yang, F., Lin, J.: Some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes. Linear Algebra Appl. 475, 186–199 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chen, X., Zhu, S., Kai, X.: Two classes of new optimal asymmetric quantum codes. Int. J. Theor. Phys. 57(6), 1829–1838 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    de Almeida, A.C.A., Palazzo, R. Jr.: A concatenated [(4,1,3)] quantum convolutional code. In: Proceedings of Information Theory Workshop, pp. 28–33 (2004)Google Scholar
  19. 19.
    Huang, Y., Chen, J., Feng, C., Chen, R.: Some families of asymmetric quantum MDS codes constructed from constacyclic codes. Int. J. Theor. Phys. 57(2), 453–464 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Huffman, W.C., Pless, V.: Fundamentals of Error-Correcting Codes. University Press, Cambridge (2003)CrossRefGoogle Scholar
  21. 21.
    Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193–1197 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kai, X., Zhu, S., Tang, Y.: Quantum negacyclic codes. Phys. Rev. A 88(1), 012326 (2013)ADSCrossRefGoogle Scholar
  23. 23.
    Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2086 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892–4914 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Krishna, A., Sarwate, D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36(4), 880–884 (1990)MathSciNetCrossRefGoogle Scholar
  26. 26.
    La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(4), 042331 (2009)ADSCrossRefGoogle Scholar
  27. 27.
    La Guardia, G.G.: New quantum MDS codes. IEEE Trans. Inf. Theory 57(8), 5551–5554 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    La Guardia, G.G.: New families of asymmetric quantum BCH codes. Quantum Inf. Comput. 11(3), 239–252 (2011)MathSciNetzbMATHGoogle Scholar
  29. 29.
    La Guardia, G.G.: Asymmetric quantum Reed-Solomon and generalized Reed–Solomon codes. Quantum Inf. Process. 11(2), 591–604 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    La Guardia, G.G.: Asymmetric quantum product codes. Int. J. Quantum Inf. 10(1), 1250005 (2012)MathSciNetCrossRefGoogle Scholar
  31. 31.
    La Guardia, G.G.: On nonbinary quantum convolutional BCH codes. Quantum Inf. Comput. 12(9–10), 820–842 (2012)MathSciNetzbMATHGoogle Scholar
  32. 32.
    La Guardia, G.G.: Asymmetric quantum codes: new codes from old. Quantum Inf. Process. 12(8), 2771–2790 (2013)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    La Guardia, G.G.: On the construction of asymmetric quantum codes. Int. J. Theor. Phys. 53(7), 2312–2322 (2014)MathSciNetzbMATHGoogle Scholar
  34. 34.
    La Guardia, G.G.: On classical and quantum MDS-convolutional BCH codes. IEEE Trans. Inf. Theory 60(1), 304–312 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    La Guardia, G.G.: On MDS-convolutional codes. Linear Algebra Appl. 448, 85–96 (2014)MathSciNetCrossRefGoogle Scholar
  36. 36.
    La Guardia, G.G.: On optimal constacyclic codes. Linear Algebra Appl. 496, 594–610 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Leng, R.G., Ma, Z.: Constructions of new families of nonbinary asymmetric quantum BCH codes and subsystem BCH codes. Sci. China Phys. Mech. 55(3), 465–469 (2012)CrossRefGoogle Scholar
  38. 38.
    Li, F., Yue, Q.: New quantum MDS-convolutional codes derived from constacyclic codes. Mod. Phys. Lett. B 29, 1550252 (2015)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Li, R., Xu, G., Guo, L.: On two problems of asymmetric quantum codes. Int. J. Mod. Phys. B 28(6), 1450017 (2013)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Lü, L., Ma, W., Li, R., Ma, Y., Guo, L.: New quantum MDS codes constructed from constacyclic codes. arXiv:1803.07927
  41. 41.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-correcting Codes. North-Holland, Amsterdam (1977)zbMATHGoogle Scholar
  42. 42.
    Qian, J., Zhang, L.: New optimal asymmetric quantum codes. Mod. Phys. Lett. B 27(2), 1350010 (2013)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Qian, J., Zhang, L.: Improved constructions for quantum maximum distance separable codes. Quantum Inf. Process. 16(1), 20 (2017)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Steane, A.M.: Enlargement of Calderbank–Shor–Steane quantum codes. IEEE Trans. Inf. Theory 45(7), 2492–2495 (1999)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf. Process. 14(3), 881–889 (2015)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Wang, L., Zhu, S.: On the construction of optimal asymmetric quantum codes. Int. J. Quantum Inf. 12(3), 1450017 (2014)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Xu, G., Li, R., Guo, L., Lü, L.: New optimal asymmetric quantum codes constructed from constacyclic codes. Int. J. Mod. Phys. B 31(5), 1750030 (2017)ADSMathSciNetCrossRefGoogle Scholar
  48. 48.
    Yan, T., Huang, X., Tang, Y.: Quantum convolutional codes derived from constacyclic codes. Mod. Phys. Lett. B 28(31), 1450241 (2014)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    Zhang, G., Chen, B., Li, L.: New optimal asymmetric quantum codes from constacyclic codes. Mod. Phys. Lett. B 28(15), 1450126 (2014)ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    Zhang, G., Chen, B., Li, L.: A construction of MDS quantum convolutional codes. Int. J. Theor. Phys. 54(9), 3182–3194 (2015)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Zhu, S., Wang, L., Kai, X.: New optimal quantum convolutional codes. Int. J. Quantum Inf. 13(3), 1550019 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Computer and Information SciencesFujian Agriculture and Forestry UniversityFuzhouChina
  2. 2.State Key Laboratory of Information Engineering in Surveying, Mapping and Remote SensingWuhan UniversityWuhanChina
  3. 3.Department of Network EngineeringChengdu University of Information TechnologyChengduChina

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