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New Quantum Codes from the Images of Cyclic Codes

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Abstract

In this paper, cyclic codes over \(\varvec{F_{q^{2m}}}\) of lengths \(\varvec{2(q^{2m}-1)}\) and \(\varvec{\frac{q^{2m}-1}{r(q-1)}}\) are considered, where \(\varvec{r\mid (q+1)}\) and \(\varvec{r\ne q+1}\). Based on the Hermitian construction, the \(\varvec{q^{2}}\)-ary images of these cyclic codes are employed to construct quantum codes. The quantum codes obtained by our technique have higher code rate than the currently known ones.

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References

  1. Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Primitive quantum BCH codes over finite fields. In: Proceedings of IEEE International Symposium on Information Theory, pp. 1114–1118 (2006)

  2. 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 

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

    Article  MathSciNet  Google Scholar 

  4. Bierbrauer, J., Edel, Y.: Quantum twisted codes. J. Comb. Des. 8, 174–188 (2000)

    Article  MathSciNet  Google Scholar 

  5. 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), 1369–1389 (1998)

    Article  MathSciNet  Google Scholar 

  6. Chen, B., Ling, S., Zhang, G.: Application of constacyclic codes to quantum MDS codes. IEEE Trans. Inf. Theory 61(3), 1474–1484 (2015)

    Article  MathSciNet  Google Scholar 

  7. Chen, H., Ling, S., Xing, C.: Quantum codes from concatenated algebraic-geometric codes. IEEE Trans. Inf. Theory 51(8), 2915–2920 (2005)

    Article  MathSciNet  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  MathSciNet  ADS  Google Scholar 

  9. Cohen, G., Encheva, S., Litsyn, S.: On binary constructions of quantum codes. IEEE Trans. Inf. Theory 45(7), 2495–2498 (1999)

    Article  MathSciNet  Google Scholar 

  10. Edel, Y.: Some good quantum twisted codes. [Online] https://www.mathi.uni-heidelberg.de/~yves/Matritzen/QTBCH/QTBCHIndex.html. Accessed on Oct 2023

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

    Article  MathSciNet  Google Scholar 

  12. Grassl, M., Geiselmann, W., Beth, T.: Quantum Reed-Solomon codes. In: Proceedings of AAECC, vol. 13, pp. 231–244 (1999)

  13. Kai, X., Li, P., Zhu, S.: Construction of quantum negacyclic BCH codes. Int. J. Quantum Inf. 16(7), 1850059 (2018)

    Article  MathSciNet  Google Scholar 

  14. Kai, X., Zhu, S.: New quantum MDS codes from negacyclic codes. IEEE Trans. Inf. Theory 59(2), 1193–1197 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. La Guardia, G.G.: Constructions of new families of nonbinary quantum codes. Phys. Rev. A 80(10), 042331(1-11) (2009)

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Li, R., Wang, J., Liu, Y., Guo, G.: New quantum constacyclic codes. Quantum Inf. Process. 18, 1–23 (2019)

    MathSciNet  ADS  Google Scholar 

  22. Lin, X.: Quantum cyclic and constacyclic codes. IEEE Trans. Inf. Theory 50(3), 547–549 (2004)

    Article  MathSciNet  Google Scholar 

  23. Ling, S., Luo, J., Xing, C.: Generalization of Steane’s enlargement construction of quantum codes and applications. IEEE Trans. Inf. Theory 56(8), 4080–4084 (2010)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  CAS  PubMed  ADS  Google Scholar 

  28. 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 

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

    Article  Google Scholar 

  30. Verma, R., Prakash, O., Islam, H., Singh, A.: New non-binary quantum codes from skew constacyclic and additive skew constacyclic codes. Eur. Phys. J. Plus. 137, 213 (2022)

    Article  Google Scholar 

  31. Wang, L., Sun, Z., Zhu, S.: Hermitian dual-containing narrow-sense constacyclic BCH codes and quantum codes. Quantum Inf. Process. 18(10), 1–40 (2019)

    Article  MathSciNet  Google Scholar 

  32. Zhu, S., Guo, H., Kai, X., Sun, Z.: New quantum codes derived from images of cyclic codes. Quantum Inf. Process. 21, 254 (2022)

    Article  MathSciNet  CAS  ADS  Google Scholar 

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Acknowledgements

This study is supported by the National Natural Science Foundation of China (No.61972126, No.62002093, No.U21A20428, No.12171134)

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Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230601, Anhui, China

    Ni Bai & Xiaoshan Kai

  2. School of Mathematics and Statistics, Hefei Normal University, Hefei, 230601, Anhui, China

    Yongsheng Tang

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  1. Ni Bai
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  2. Xiaoshan Kai
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  3. Yongsheng Tang
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These authors contributed equally to this work.

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Correspondence to Ni Bai.

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Bai, N., Kai, X. & Tang, Y. New Quantum Codes from the Images of Cyclic Codes. Int J Theor Phys 63, 2 (2024). https://doi.org/10.1007/s10773-023-05531-5

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