Skip to main content
SpringerLink
Log in

Optimal quaternary Hermitian LCD codes and their related codes

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

Abstract

Linear codes with complementary dual (LCD codes) play an important role in armoring implementations against side-channel attacks and fault injection attacks. Hermitian LCD codes are a special class of LCD codes, which were proved to be asymptotically good. In this paper, several classes of optimal quaternary Hermitian LCD codes are constructed via constacyclic codes. The binary images of these classes of quaternary Hermitian LCD codes are characterized. Several classes of optimal binary LCD codes with minimum distance 6, and optimal binary linear codes with one-dimensional hull and minimum distance 6 are also presented.

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

Code availibility

Not applicable.

References

  1. Araya M., Harada M.: On the classification of quaternary optimal Hermitian LCD codes. Cryptogr. Commun. 14, 833–847 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  2. Araya M., Harada M., Saito K.: Quaternary Hermitian linear complementary dual codes. IEEE Trans. Inf. Theory 66(5), 2751–2759 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  3. Assmus E.F. Jr., Key J.D.: Affine and projective planes. Discret. Math. 83(2–3), 161–187 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  4. Aydin N., Siap I., Ray-Chaudhuri D.K.: The structure of 1-generator quasi-twisted codes and new linear codes. Des. Codes Cryptogr. 24(3), 313–326 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  5. Boonniyoma K., Jitman S.: Complementary dual subfield linear codes over finite fields. arxiv:1605.06827

  6. Bouyuklieva S.: Optimal binary LCD codes. Des. Codes Cryptogr. 89(11), 2445–2461 (2021).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Carlet C., Mesnager S., Tang C., 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  MATH  Google Scholar 

  9. Carlet C., Mesnager S., Tang C., Qi Y.: New characterization and parametrization of LCD codes. IEEE Trans. Inf. Theory 65(1), 39–49 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  10. Carlet C., Li C., Mesnager S.: Linear codes with small hulls in semi-primitive case. Des. Codes Cryptogr. 87(12), 3063–3075 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  11. Dinh H.Q.: Repeated-root cyclic and negacyclic codes of length \(6p^s\). Contemp. Math. 609, 69–87 (2014).

    Article  MATH  Google Scholar 

  12. Fang W., Wen J., Fu F.: A \(q\)-polynomial approach to constacyclic codes. Finite Fields Appl. 47, 161–182 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  13. Fu Q., Li R., Fu F., Rao Y.: On the construction of binary optimal LCD codes with short length. Int. J. Found. Comput. Sci. 30(08), 1237–1245 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  14. Galindo C., Geil O., Hernando F., Ruano D.: New binary and ternary LCD codes. IEEE Trans. Inf. Theory 65(2), 1008–1016 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  15. Galvez L., Kim J.L., Lee N., Roe Y.G., Won B.S.: Some bounds on binary LCD codes. Cryptogr. Commun. 10(4), 719–728 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  16. Grassl M.: Bounds on the minimum distance of linear codes and quantum codes. http://www.codetables.de.

  17. Guenda K., Jitman S., Gulliver T.A.: Constructions of good entanglement-assisted quantum error correcting codes. Des. Codes Cryptogr. 86(1), 121–136 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  18. Güneri C., Özkaya B., Solé P.: Quasi-cyclic complementary dual codes. Finite Fields Appl. 42, 67–80 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  19. Harada M.: Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes. Des. Codes Cryptogr. 89(10), 2295–2312 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  20. Huang X., Yue Q., Wu Y., Shi X., Michel J.: Binary primitive LCD BCH codes. Des. Codes Cryptogr. 88(12), 2453–2473 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  21. Huffman W., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge Univ. Press, Cambridge (2010).

    MATH  Google Scholar 

  22. Ishizuka K., Saito K.: On the existence of quaternary Hermitian LCD codes with Hermitian dual distance \(1\). Discret. Math. 345(2), 112702 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  23. Krishna A., Sarwate D.V.: Pseudocyclic maximum-distance-separable codes. IEEE Trans. Inf. Theory 36(4), 880–884 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  24. Leon J.: Computing automorphism groups of error-correcting codes. IEEE Trans. Inf. Theory 28(3), 496–511 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  25. Leon J.: Permutation group algorithms based on partitions, I: theory and algorithms. J. Symbolic Comput. 12(4–5), 533–583 (1991).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Li C., Zeng P.: Constructions of linear codes with one-dimensional hull. IEEE Trans. Inf. Theory 65(3), 1668–1676 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  28. Li C., Ding C., Li S.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63(7), 4344–4356 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  29. 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 

  30. Liu Z., Wang J.: Further results on Euclidean and Hermitian linear complementary dual codes. Finite Fields Appl. 59, 104–133 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  31. Liu Y., Li R., Fu Q., Lv L., Rao Y.: Some binary BCH codes with length \(2^m+1\). Finite Fields Appl. 55, 109–133 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  32. Lu L., Zhan X., Yang S., Cao H.: Optimal quaternary Hermitian LCD codes. arXiv:2010.10166.

  33. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1997).

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Mesnager S., Tang C.: Fast algebraic immunity of Boolean functions and LCD codes. IEEE Trans. Inf. Theory 67(7), 4828–4837 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  36. Qian L., Cao X., Mesnager S.: Linear codes with one-dimensional hull associated with Gaussian sums. Cryptogr. Commun. 13, 225–243 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  37. Qian L., Cao X., Lu W., Solé P.: A new method for constructing linear codes with small hulls. Des. Codes Cryptogr. (2021). https://doi.org/10.1007/s10623-021-00940-1.

    Article  MATH  Google Scholar 

  38. Rao Y., Li R., Lv L., Chen G., Zuo F.: On binary LCD cyclic codes. Procedia Comput. Sci. 107, 778–783 (2017).

    Article  Google Scholar 

  39. Sendrier N.: Finding the permutation between equivalent linear codes: the support splitting algorithm. IEEE Trans. Inf. Theory 46(4), 1193–1203 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  40. Sendrier N., Skersys G.: On the computation of the automorphism group of a linear code. In: Proceedings of IEEE ISIT’2001, vol. 13, Washington, DC (2001).

  41. Shi M., Özbudak F., Xu L., Solé P.: LCD codes from tridiagonal Toeplitz matrices. Finite Fields Appl. 75, 101892 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  42. Sok L.: On Hermitian LCD codes and their Gray image. Finite Fields Appl. 62, 101623 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  43. Sok L.: A new construction of linear codes with one-dimensional hull. Des. Codes Cryptogr. (2022). https://doi.org/10.1007/s10623-021-00991-4.

    Article  MathSciNet  MATH  Google Scholar 

  44. Sok L.: On linear codes with one-dimensional Euclidean hull and their applications to EAQECCs. IEEE Trans. Inf. Theory 68(7), 4329–4343 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  45. Sok L.: MDS linear codes with one-dimensional hull. Cryptogr. Commun. 14, 949–971 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  46. Sok L., Shi M., Solé P.: Constructions of optimal LCD codes over large finite fields. Finite Fields Appl. 50, 138–153 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  47. Wang Y., Tao R.: Constructions of linear codes with small hulls from association schemes. Adv. Math. Commun. 16(2), 349–364 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  48. Wu Y.: Twisted Reed–Solomon codes with one-dimensional hull. IEEE Commun. Lett. 25(2), 383–386 (2020).

    Article  Google Scholar 

  49. Wu Y., Lee Y.: Binary LCD codes and self-orthogonal codes via simplicial complexes. IEEE Comnun. Lett. 24(6), 1159–1162 (2020).

    Article  Google Scholar 

  50. Yang Y., Cai W.: On self-dual constacyclic codes over finite fields. Des. Codes Cryptogr. 74(2), 355–364 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  51. Zhan X., Li R., Lu L., Li H.: Quaternary Hermitian linear complementary dual codes with small distance. in: Processing of IEEE ICISE-IE, Sanya, China (2020).

  52. Zhou Z., Li X., Tang C., Ding C.: Binary LCD codes and self-orthogonal codes from a generic construction. IEEE Trans. Inf. Theory 65(1), 16–27 (2019).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the reviewers and the Editor, Prof. Tor Helleseth, for their detailed comments and suggestions that highly improved the presentation of this article. The first author also wishes to thank Prof. Cunsheng Ding for helpful discussion. This work was supported by National Natural Science Foundation of China under Grant Nos. 62002093, 61972126, U21A20428, and 12171134. All the code examples in this paper were computed with the Magma software package.

Author information

Authors and Affiliations

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

    Zhonghua Sun, Sujuan Huang & Shixin Zhu

Authors
  1. Zhonghua Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sujuan Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shixin Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhonghua Sun.

Ethics declarations

Conflict of interest

The authors declare there are no conflicts of interest.

Additional information

Communicated by T. Helleseth.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, Z., Huang, S. & Zhu, S. Optimal quaternary Hermitian LCD codes and their related codes. Des. Codes Cryptogr. 91, 1527–1558 (2023). https://doi.org/10.1007/s10623-022-01166-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-022-01166-5

Keywords

Mathematics Subject Classification

Search

Navigation