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Linear \(\ell \)-intersection pairs of codes and their applications

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Abstract

In this paper, a linear \(\ell \)-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear \(\ell \)-intersection pair if their intersection has dimension \(\ell \). Characterizations and constructions of such pairs of codes are given in terms of the corresponding generator and parity-check matrices. Linear \(\ell \)-intersection pairs of MDS codes over \({\mathbb {F}}_q\) of length up to \(q+1\) are given for all possible parameters. As an application, linear \(\ell \)-intersection pairs of codes are used to construct entanglement-assisted quantum error correcting codes. This provides a large number of new MDS entanglement-assisted quantum error correcting codes.

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References

  1. Bosma W., Cannon J., Playoust C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Brun T., Devetak I., Hsieh M.H.: Catalytic quantum error correction. IEEE Trans. Inf. Theory 60, 3073–3089 (2014).

    Article  MathSciNet  Google Scholar 

  4. Carlet C., Guilley S.: Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun. 10, 131–150 (2016).

    Article  MathSciNet  Google Scholar 

  5. Carlet C., Güneri C., Mesnager S., Özbudak F.: Construction of some codes suitable for both side channel and fault injection attacks. In: Proceedings of International Workshop on the Arithmetic of Finite Fields (WAIFI 2018), Bergen (2018).

    Chapter  Google Scholar 

  6. Carlet C., Güneri C., Özbudak F., Özkaya B., Solé P.: On linear complementary pairs of codes. IEEE Trans. Inf. Theory 64, 6583–6589 (2018).

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Chen J., Huang Y., Feng C., Chen R.: Entanglement-assisted quantum MDS codes constructed from negacyclic codes. Quantum Inf. Process 16, 303 (2017).

    Article  MathSciNet  Google Scholar 

  9. Fan Y., Zhang L.: Galois self-dual constacyclic codes. Des. Codes Cryptogr. 84, 473–492 (2017).

    Article  MathSciNet  Google Scholar 

  10. Gauss C.F.: Untersuchungen Über Höhere Arithmetik, 2nd edn. Chelsea, New York (1981).

    Google Scholar 

  11. Grassl, M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de. Accessed 20 June 2019.

  12. 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 

  13. Hsich M.H., Devetak I., Brun T.: General entanglement-assisted quantum error-correcting codes. Phys. Rev. A 76, 062313 (2007).

    Article  Google Scholar 

  14. Jin L.: Construction of MDS codes with complementary duals. IEEE Trans. Inf. Theory 63, 2843–2847 (2017).

    MathSciNet  MATH  Google Scholar 

  15. Lacan J., Fimes J.: Systematic MDS erasure codes based on Vandermonde matrices. IEEE Commun. Lett. 8, 570–572 (2004).

    Article  Google Scholar 

  16. Lu L., Li R., Guo L., Ma R., Liu Y.: Entanglement-assisted quantum MDS codes from negacyclic codes. Quantum Inf. Process 17, 69 (2018).

    Article  MathSciNet  Google Scholar 

  17. Lu L., Ma W., Li R., Ma Y., Liu Y., Cao H.: Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance. Finite Fields Appl. 53, 309–325 (2018).

    Article  MathSciNet  Google Scholar 

  18. Luo G., Cao X., Chen X.: MDS codes with hulls of arbitrary dimensions and their quantum error correction. IEEE Trans. Inf. Theory 65, 2944–2952 (2019).

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  20. Massey J.L.: Linear codes with complementary duals. Discret. Math. 106–107, 337–342 (1992).

    Article  MathSciNet  Google Scholar 

  21. Qian J., Zhang L.: On MDS linear complementary dual codes and entanglement-assisted quantum codes. Des. Codes Cryptogr. 86, 1565–1572 (2018).

    Article  MathSciNet  Google Scholar 

  22. Roth R.M., Lempel A.: On MDS codes via Cauchy matrices. IEEE Trans. Inf. Theory 35, 1314–1319 (1989).

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for very helpful comments. S. Jitman was supported by the Thailand Research Fund and Silpakorn University under Research Grant RSA6280042.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Victoria, PO Box 1700, STN CSC, Victoria, BC, V8W 2Y2, Canada

    Kenza Guenda & T. Aaron Gulliver

  2. Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom, 73000, Thailand

    Somphong Jitman & Satanan Thipworawimon

Authors
  1. Kenza Guenda
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  2. T. Aaron Gulliver
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  3. Somphong Jitman
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  4. Satanan Thipworawimon
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Correspondence to Somphong Jitman.

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Communicated by V. D. Tonchev.

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Guenda, K., Gulliver, T.A., Jitman, S. et al. Linear \(\ell \)-intersection pairs of codes and their applications. Des. Codes Cryptogr. 88, 133–152 (2020). https://doi.org/10.1007/s10623-019-00676-z

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  • DOI: https://doi.org/10.1007/s10623-019-00676-z

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