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New quantum codes derived from images of cyclic codes

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Abstract

Let q be a prime power and \(m \ge 2\) be a positive integer. Based on classical cyclic codes, we construct new quantum codes by using images of cyclic codes. Three classes of quantum codes are derived from the \({\mathbb {F}}_{q^2}\) images of cyclic codes over \({\mathbb {F}}_{q^{2m}}\). Compared with the known ones, these new quantum codes have larger minimum distance or higher code rate.

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References

  1. Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), 2493–2496 (1995)

    Article  ADS  Google Scholar 

  2. Steane, A.M.: Multiple particle interference and quantum error correction. Proc. R. Soc. London A 452(1), 2552–2577 (1996)

    ADS  MathSciNet  MATH  Google Scholar 

  3. Steane, A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77(5), 793–797 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  4. Calderbank, A.R., Rainds, E.M., Shor, P.M., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44(4), 1369–1389 (1998)

    Article  MathSciNet  Google Scholar 

  5. Feng, K.Q., Chen, H.: Quantum Error-Correcting Codes. Science Press, Beijing (2010)

    Google Scholar 

  6. Ashikhmin, A.R., Knill, E.: Nonbinary quantum stablizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  8. Christensen, R.B., Geil, O.: On steane-enlargement of quantum codes from Cartesian product point sets. Quantum Inf. Process. 19(7), 192 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  9. Li, R.H., Wang, J.L., Liu, Y., Guo, G.M.: New quantum constacyclic codes. Quantum Inf. Process. 18(5), 127 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  10. Kai, X.S., Zhu, S.X., Li, P.: Constacyclic codes and some new quantum MDS codes. IEEE Trans. Inf. Theory 60(4), 2080–2086 (2014)

    Article  MathSciNet  Google Scholar 

  11. Zhu, S.X., Jiang, W., Chen, X.J.: New entanglement-assisted quantum MDS codes with length \((q^2+1)/5\). Quantum Inf. Process 19(7), 211 (2020)

    Article  ADS  Google Scholar 

  12. Chen, X.J., Zhu, S.X., Jiang, W., Pang, B.B.: Four classes of new entanglement-assisted quantum optimal codes. J. Appl. Math. Comput. (2021). https://doi.org/10.1007/s12190-021-01523-y

    Article  MathSciNet  MATH  Google Scholar 

  13. Va Guardia, G.G.: Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes. Quantum Inf. Process. 11(2), 591–604 (2012)

    Article  MathSciNet  Google Scholar 

  14. Luo, G.J., Cao, X.W.: Two new families of entanglement-assisted quantum MDS codes from generalized Reed-Solomon codes. Quantum Inf. Process. 18(89), 1–12 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. IEEE Int. Symp. Inf. Theory, arXiv:1810.07004 (2006)

  16. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Int. Symp. Inf. Theory 53(3), 1183–1188 (2007)

    Article  MathSciNet  Google Scholar 

  17. Wang, J.L., Li, R.H., Lv, J.J., Song, H.: Entanglement-assisted quantum codes from cyclic codes and negacyclic codes. Quantum Inf. Process. 19(5), 138 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  18. Thangaraj, A., McLaughin, S.W.: Quantum codes from cyclic codes over \(GF(4^m)\). IEEE Trans. Inf. Theory 47(3), 1176–1178 (2001)

    Article  Google Scholar 

  19. Seguin, G.E.: The \(q\)-ary image of a \(q^m\)-ary cyclic code. IEEE Trans. Inf. Theory 41(2), 387–399 (1995)

    Article  Google Scholar 

  20. Sundeep, B., Thangaraj, A.: Self-orthogonality of \(q\)-ary images of \(q^m\)-ary codes and quantum code construction. IEEE Trans. Inf. Theory 53(7), 2480–2489 (2007)

    Article  Google Scholar 

  21. Kai, X.S., Zhu, S.X., Sun, Z.H.: The images of constacyclic codes and new quantum codes. Quantum Inf. Process. 19(7), 212 (2020)

    Article  ADS  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  23. Wan, Z.X.: Lectures on Finite Fields and Galois Rings. World Scientific, Singapore (2003)

    Book  Google Scholar 

  24. Edel, Y.: Some good quantum twisted codes. https://www.mathi.uniheidelberg.de/yves/Matritzen/QTBCH/QTBCHIndex.html (2020)

  25. Feng, K.Q., Ling, S., Xing, C.P.: Asymptotic bounds on quantum codes from algebraic geometry codes. IEEE Trans. Inf. Theory 52(3), 986–991 (2006)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This research is supported by the National Natural Science Foundation of China (Nos. 61772168, 61972126, 12001002, 62002093) and the Natural Science Foundation of Anhui Province (No. 2008085QA04).

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Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230009, Anhui, People’s Republic of China

    Shixin Zhu, Hongzhe Guo, Xiaoshan Kai & Zhonghua Sun

  2. Intelligent Interconnected Systems Laboratory of Anhui Province (Hefei University of Technology), Hefei, 230009, Anhui, People’s Republic of China

    Shixin Zhu, Hongzhe Guo, Xiaoshan Kai & Zhonghua Sun

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  1. Shixin Zhu
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  2. Hongzhe Guo
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  3. Xiaoshan Kai
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  4. Zhonghua Sun
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Correspondence to Hongzhe Guo.

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This research is supported by the National Natural Science Foundation of China (Nos. 61772168, 61972126, 12001002, 62002093) and the Natural Science Foundation of Anhui Province (No.2008085QA04)

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Zhu, S., Guo, H., Kai, X. et al. New quantum codes derived from images of cyclic codes. Quantum Inf Process 21, 254 (2022). https://doi.org/10.1007/s11128-022-03603-9

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