Skip to main content
SpringerLink
Log in

Binary primitive LCD BCH codes

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

Linear complementary dual (LCD) codes have attracted much attention in recent years due to their applications in implementations against side-channel attacks and fault injection attacks. Comparing coset leaders, we introduce the new concept of absolute coset leaders, which provides advantages for constructing LCD BCH codes. We then give explicit presentations for the largest, second largest and third largest absolute coset leaders and use them to construct binary LCD BCH codes. Lastly, we determine the parameters of these LCD BCH codes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Carlet C., Guilley S.: Complementary dual codes for countermeasures to side-channel attacks. Adv. Math. Commun. 10(1), 131–150 (2016).

    Article  MathSciNet  Google Scholar 

  2. 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 

  3. Carlet C., Mesnager S., Tang C., Qi Y., Pellikaan R.: Linear codes over \(F_q\) which are equivalent to LCD codes. IEEE Trans. Inf. Theory 64, 3010–3017 (2018).

    Article  Google Scholar 

  4. Chen B., Liu H.: New constructions of MDS codes with complementary duals. IEEE Trans. Inf. Theory 63, 2843–2847 (2017).

    Google Scholar 

  5. Delsarte P.: On subfield subcodes of modified Reed–Solomon codes. IEEE Trans. Inf. Theory 21, 575–576 (1975).

    Article  MathSciNet  Google Scholar 

  6. Ding C.: Parameters of several classes of BCH codes. IEEE Trans. Inf. Theory 61, 5322–5330 (2015).

    Article  MathSciNet  Google Scholar 

  7. Ding C., Fan C., Zhou Z.: The dimension and minimum distance of two classes of primitive BCH codes. Finite Fields Appl. 45, 237–263 (2017).

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Li C.: Hermitian LCD codes from cyclic codes. Des. Codes Cryptogr. 86, 2261–2278 (2018).

    Article  MathSciNet  Google Scholar 

  10. Li C., Yue Q., Li F.: Weight distributions of cyclic codes with respect to pairwise coprime order elements. Finite Fields Appl. 28, 94–114 (2014).

    Article  MathSciNet  Google Scholar 

  11. Li C., Ding C., Li S.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63, 4344–4356 (2017).

    Article  MathSciNet  Google Scholar 

  12. Li S., Ding C., Xiong M., Ge G.: Narrow-sense BCH codes over GF\((q)\) with length \(n=\frac{q^m-1}{q-1}\). IEEE Trans. Inf. Theory 63, 7219–7236 (2017).

    Article  Google Scholar 

  13. Li S., Li C., Ding C., Liu H.: Two families of LCD BCH codes. IEEE Trans. Inf. Theory 63(9), 5699–5717 (2017).

    MathSciNet  MATH  Google Scholar 

  14. Li C., Wu P., Liu F.: On two classes of primitive BCH codes and some related codes. IEEE Trans. Inf. Theory 65, 3830–3840 (2019).

    Article  MathSciNet  Google Scholar 

  15. Li F., Yue Q., Wu Y.: Designed distances and parameters of new LCD BCH codes over finite fields. Cryptogr. Commun. (2019). https://doi.org/10.1007/s12095-019-00385-3.

    Article  MATH  Google Scholar 

  16. Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (2008).

    MATH  Google Scholar 

  17. Liu H., Ding C., Li C.: Dimensions of three types of BCH codes over GF\((q)\). Discret. Math. 340, 1910–1927 (2017).

    Article  MathSciNet  Google Scholar 

  18. Massey J.L.: Reversible codes. Inf. Control 7(3), 369–380 (1964).

    Article  MathSciNet  Google Scholar 

  19. Myerson G.: Period polynomials and Gauss sums for finite fields. Acta Arith. 39, 251–264 (1981).

    Article  MathSciNet  Google Scholar 

  20. Shi X., Yue Q., Yang S.: New LCD MDS codes constructed from generalized Reed–Solomon codes. J. Algebra Appl. 18, 1950150 (2018).

    Article  MathSciNet  Google Scholar 

  21. Tzeng K.K., Hartmann C.R.P.: On the minimum distance of certain reversible cyclic codes. IEEE Trans. Inf. Theory 16(5), 644–646 (1970).

    Article  MathSciNet  Google Scholar 

  22. Yan H., Liu H., Li C., Yang S.: Parameters of LCD BCH codes with two lengths. Adv. Math. Commun. 12, 579–594 (2018).

    Article  MathSciNet  Google Scholar 

  23. Yang X., Massey J.L.: The necessary and sufficient condition for a cyclic code to have a complementary dual. Discret. Math. 126(1), 391–393 (1994).

    Article  Google Scholar 

  24. Zhou Z., Li X., Tang C., Ding C.: Binary LCD codes and self-orthogonal codes from a generic construction. IEEE Trans. Inf. Theory 65, 16–27 (2019).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research was supported by the National Natural Science Foundation of China (No. 61772015) and the Foundation of Jinling Institute of Technology (No. JIT-B-202016). The authors are very thankful to the reviewers and the editor for their valuable comments and suggestions to this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Jinling Institute of Technology, Nanjing, 211169, People’s Republic of China

    Xinmei Huang

  2. State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, People’s Republic of China

    Xinmei Huang & Qin Yue

  3. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, People’s Republic of China

    Qin Yue

  4. Department of Mathematics, Ewha Womans University, Seoul, 03760, South Korea

    Yansheng Wu

  5. Department of Mathematics, Nanjing Forestry University, Nanjing, 210037, People’s Republic of China

    Xiaoping Shi

  6. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, People’s Republic of China

    Jerod Michel

Authors
  1. Xinmei Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qin Yue
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yansheng Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiaoping Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jerod Michel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinmei Huang.

Additional information

Communicated by C. Ding.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, X., Yue, Q., Wu, Y. et al. Binary primitive LCD BCH codes. Des. Codes Cryptogr. 88, 2453–2473 (2020). https://doi.org/10.1007/s10623-020-00795-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-020-00795-y

Keywords

Mathematics Subject Classification

Search

Navigation