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A class of constacyclic BCH codes and new quantum codes

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Abstract

Constacyclic BCH codes have been widely studied in the literature and have been used to construct quantum codes in latest years. However, for the class of quantum codes of length \(n=q^{2m}+1\) over \(F_{q^2}\) with q an odd prime power, there are only the ones of distance \(\delta \le 2q^2\) are obtained in the literature. In this paper, by a detailed analysis of properties of \(q^{2}\)-ary cyclotomic cosets, maximum designed distance \(\delta _\mathrm{{max}}\) of a class of Hermitian dual-containing constacyclic BCH codes with length \(n=q^{2m}+1\) are determined, this class of constacyclic codes has some characteristic analog to that of primitive BCH codes over \(F_{q^2}\). Then we can obtain a sequence of dual-containing constacyclic codes of designed distances \(2q^2<\delta \le \delta _\mathrm{{max}}\). Consequently, new quantum codes with distance \(d > 2q^2\) can be constructed from these dual-containing codes via Hermitian Construction. These newly obtained quantum codes have better code rate compared with those constructed from primitive BCH codes.

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References

  1. Shor, P.W.: Scheme for reducing decoherence in quantum computing memory. Phys. Rev. A 52, R2493 (1995)

    Article  ADS  Google Scholar 

  2. Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. 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 

  4. Gottesman, D.: Stabilizer codes and quantum error correction. Ph.D. Thesis, California Institute of Technology (1997)

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

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

  7. Rains, E.M.: Non-binary quantum codes. IEEE. Trans. Inf. Theory 45, 1827–1832 (1999)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Ling, S., Luo, J., Xing, C.: Generalization of Steane’s enlargement construction of quantum codes and applications. IEEE Trans. Inf. Theory 56, 4080–4084 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Grassl, M., Beth, T.: Quantum BCH codes. Proc. X. int’l. Symp. Theoretical. Electrical Engineering Magdeburg, 207-212 (1999)

  13. Li, R., Li, X.: Quantum codes constructed from binary cyclic codes. Int. J. Quantum Inf. 2, 265–272 (2004)

    Article  MATH  Google Scholar 

  14. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. Proc. Int. Symp. Inf. Theory, 1114-1118 (2006)

  15. 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 

  16. Guardia, G.G.La: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80, 042331 (2009)

    Article  ADS  Google Scholar 

  17. Li, R., Zuo, F., Liu, Y.: A study of skew symmetric \(q^2\)-cyclotomic coset and its application. J. Air Force Eng. Univ. 12(1), 87–89 (2011)

    Google Scholar 

  18. 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 

  19. Kai, X., Zhu, S.: Quantum negacyclic codes. Phys. Rev. A 88, 012326 (2013)

    Article  ADS  Google Scholar 

  20. Kai, X., Zhu, S., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60, 2080–2086 (2014)

    Article  MathSciNet  Google Scholar 

  21. Hu, X., Zhang, G., Chen, B.: Constructions of new nonbinary quantum codes. Int. J. Theory Phys. 54, 92–99 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61, 1474–1484 (2015)

    Article  MathSciNet  Google Scholar 

  23. Guardia, G.G.La: On optimal constacyclic codes. Linear Algebra Appl. 496, 594–610 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, L., Zhu, S.: New quantum MDS codes derived from constacyclic codes. Quantum Inf. Processing 14 3, 881-889(2015). See also arXiv:1405.5421v1

  25. Zhang, T., Ge, G.: Some new class of quantum MDS codes from constacyclic codes. IEEE Trans. Inf. Theory 61, 5224–5228 (2015)

    Article  Google Scholar 

  26. 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 

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

    Article  MATH  Google Scholar 

  28. Peterson, W.W., Weldon, E.J.: Error-correcting codes. The M.I.T. Press, Cambridge (1972)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  32. Sloane, N.J.A., Thompson, J.G.: Cyclic self-dual codes. IEEE Trans. Inf. Theory 29, 364–366 (1983)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China under Grant No.11471011 and Natural Science Foundation of Shaanxi under Grant No.2015JM1023.

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Authors and Affiliations

  1. School of Science, Air Force Engineering University, Xi’an, 710051, Shaanxi, People’s Republic of China

    Yang liu, Ruihu Li, Liangdong Lv & Yuena Ma

  2. The First Aeronautical College of Air Force, Xinyang, 464000, Henan, People’s Republic of China

    Yang liu

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  1. Yang liu
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  2. Ruihu Li
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  3. Liangdong Lv
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  4. Yuena Ma
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Correspondence to Ruihu Li.

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liu, Y., Li, R., Lv, L. et al. A class of constacyclic BCH codes and new quantum codes. Quantum Inf Process 16, 66 (2017). https://doi.org/10.1007/s11128-017-1533-y

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