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\(\sigma \)-LCD codes over finite chain rings

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Abstract

In this work, we first generalize the \(\sigma \)-LCD codes over finite fields to \(\sigma \)-LCD codes over finite chain rings. Under suitable conditions, linear codes over finite chain rings that are \(\sigma \)-LCD codes are characterized. Then we provide a necessary and sufficient condition for free constacyclic codes over finite chain rings to be \(\sigma \)-LCD. We also get some new binary LCD codes of different lengths which come from Gray images of constacyclic \(\sigma \)-LCD codes over \(\mathbb {F}_{2}+\gamma \mathbb {F}_{2}+\gamma ^2\mathbb {F}_{2}\). Finally, for special finite chain rings \(\mathbb {F}_{q}+\gamma \mathbb {F}_{q}\), we define a new Gray map \(\Phi \) from \((\mathbb {F}_{q}+\gamma \mathbb {F}_{q})^n\) to \(\mathbb {F}_{q}^{2n}\), and by using \(\sigma \)-LCD codes over finite chain rings \(\mathbb {F}_{q}+\gamma \mathbb {F}_{q}\), we construct new entanglement-assisted quantum error-correcting (abbreviated to EAQEC) codes with maximal entanglement and parts of them are MDS EAQEC codes.

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References

  1. Alkhamees Y.: The determination of the group of automorphisms of a finite chain ring of characteristic \(p\). Q. J. Math. 42, 387–391 (1991).

    Article  MathSciNet  Google Scholar 

  2. Bhowmick, S., Fotue-Tabue, A., Martínez-Moro, E., Bandi, R., Bagchi, S.: Do non-free LCD codes over finite commutative Frobenius rings exist? arXiv:1901.10836v1 (2019).

  3. Bowen G.: Entanglement required in achieving entanglement-assisted channel capacities. Phys. Rev. A 66, 052313 (2002).

    Article  Google Scholar 

  4. Brun T.A., Devetak I., Hsieh M.H.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006).

    Article  MathSciNet  Google Scholar 

  5. Carlet, C., Guilley. S.: Complementary dual codes for counter-measures to side-channel attacks. In: Proceedings of the 4th ICMCTA Meeting, Palmela, Portugal (2014).

  6. Carlet C., Mesnager S., Tang C.M., Qi Y.: Euclidean and Hermitian LCD MDS codes. Des. Codes Cryptogr. 86(11), 2605–2618 (2018).

    Article  MathSciNet  Google Scholar 

  7. Carlet C., Mesnager S., Tang C.M., Qi Y., Pellikaan R.: Linear codes over \(\mathbb{F}_q\) are equivalent to LCD codes for \(q>3\). IEEE Trans. Inf. Theory 64(4), 3010–3017 (2018).

    Article  Google Scholar 

  8. Carlet C., Mesnager S., Tang C.M., Qi Y.: On \(\sigma \)-LCD codes. IEEE Trans. Inf. Theory 65(3), 1694–1704 (2019).

    Article  MathSciNet  Google Scholar 

  9. Dougherty S.T., Kim J.L., Kulosman H., Liu H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16, 14–26 (2010).

    Article  MathSciNet  Google Scholar 

  10. Dougherty S.T., Liu H., Park Y.H.: Lifts codes over finite chain rings. Math. J. Okayama Univ. 53, 39–53 (2011).

    MathSciNet  MATH  Google Scholar 

  11. Fan Y., Ling S., Liu H.: Matrix product codes over finite commutative Frobenius rings. Des. Codes Cryptogr. 71, 201–227 (2014).

    Article  MathSciNet  Google Scholar 

  12. Fotue-Tabue, A., Martínez-Moro, E., Blackford, T.: On polycyclic codes over a finite chain ring. Adv. Math. Commun. arXiv:1811.07975 (2018).

  13. Guenda K., Jitman S., Gulliver T.A.: Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Cryptogr. 86, 121–136 (2018).

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  15. Liu X., Liu H.: Quantum codes from linear codes over finite chain rings. Quantum Inf. Process. 16, 240 (2017).

    Article  MathSciNet  Google Scholar 

  16. Liu X., Liu H.: LCD codes over finite chain rings. Finite Fields Appl. 15, 1–19 (2015).

    Article  MathSciNet  Google Scholar 

  17. Liu X., Fan Y., Liu H.: Galois LCD codes over finite fields. Finite Fields Appl. 49, 227–242 (2018).

    Article  MathSciNet  Google Scholar 

  18. Massey J.L.: Linear codes with complementary duals. Discret. Math. 106(167), 337–342 (1992).

    Article  MathSciNet  Google Scholar 

  19. McDonald B.R.: Finite Rings with Identity. Marcel Dekker, New York (1974).

    MATH  Google Scholar 

  20. Norton G.H., Sǎlǎgean A.: On the structure of linear and cyclic codes over finite chain rings. Appl. Algebra Eng. Commun. Comput. 10, 489–506 (2000).

    Article  MathSciNet  Google Scholar 

  21. Wilde M.M., Brun T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77, 064302 (2008).

    Article  Google Scholar 

  22. Yang X., Massey J.L.: The condition for a cyclic code to have a complementary dual. Discret. Math. 126, 391–393 (1994).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for their valuable comments and suggestions, which improved the quality of the manuscript. This research was supported by Research Funds of Hubei Province (Grant No. Q20174503), and Research Project of Hubei Polytechnic University (Grant No. 17xjz03A).

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Authors and Affiliations

  1. School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, Hubei, People’s Republic of China

    Xiusheng Liu

  2. School of Science, Hubei University of Technology, Wuhan, 430068, Hubei, People’s Republic of China

    Hualu Liu

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  1. Xiusheng Liu
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  2. Hualu Liu
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Correspondence to Xiusheng Liu.

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Communicated by C. Carlet.

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Liu, X., Liu, H. \(\sigma \)-LCD codes over finite chain rings. Des. Codes Cryptogr. 88, 727–746 (2020). https://doi.org/10.1007/s10623-019-00706-w

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