Skip to main content
SpringerLink
Log in

Optimal Binary Linear Complementary Pairs of Codes

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

A pair of linear codes (C, D) of length n over \(\mathbb {F}_q\) is called a linear complementary pair (LCP) if their direct sum yields the full space \(\mathbb {F}_q^n\). By a result of Carlet et al. (2019), the best security parameters of binary LCPs of codes are left open. Motivated by this, we study binary LCPs of codes. We describe a sufficient condition for binary LCPs of codes which are not optimal. We carry out an exhaustive search to determine the best security parameters for binary LCPs of codes up to length 18. We also obtain results on optimal binary LCPs of codes for infinitely many parameters. For any \(k\ge 2\) and length n congruent to 0 or 1 mod \((2^k-1)\), we prove that binary [n, k] LCPs of codes are optimal. Binary LCPs of codes of dimensions 2, 3, and 4 are also optimal for all lengths except for two instances, when \((n,k)=(4,3)\) and (8, 4). We provide explicit constructions of these infinite families of optimal LCPs. Our results also indicate that many security parameters coming from binary LCPs of codes exceed those from binary LCD codes by 1 or 2.

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. Araya, M., Harada, M.: On the minimum weights of binary linear complementary dual codes. Cryptogr. Commun. 12, 285–300 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Araya, M., Harada, M., Saito, K.: Characterization and classification of optimal LCD codes. Des. Codes Cryptogr. 89, 617–640 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Araya, M., Harada, M., Saito, K.: On the minimum weights of binary LCD codes and ternary LCD codes. Finite Fields Appl. 76, 101925 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borello, M., de la Cruz, J., Willems, W.: A note on linear complementary pairs of group codes. Discrete Math. 343, 111905 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Bringer, J., Carlet, C., Chabanne, H., Guilley, S., Maghrebi, H.: Orthogonal direct sum masking: A smartcard friendly computation paradigm in a code, with builtin protection against side-channel and fault attacks, WISTP, Heraklion, Crete, LNCS, vol. 8501, pp. 40–56. Springer, Berlin, Heidelberg (2014)

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

  9. 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, 3010–3017 (2018)

  10. 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  MATH  Google Scholar 

  11. Dougherty, S.T., Kim, J.-L., Özkaya, B., Sok, L., Solé, P.: The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices. Int. J. Information and Coding Theory. 4, 116–128 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Güneri, C., Özkaya, B., Sayıcı, S.: On linear complementary pair of \(nD\) cyclic codes. IEEE Commun. Lett. 22, 2404–2406 (2018)

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Harada, M., Saito, K.: Binary linear complemetary dual codes. Cryptogr. Commun. 11, 677–696 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  17. Jaffe, D.: Optimal binary linear codes of length \(\le 30\). Discrete Math. 223, 135–155 (2000)

  18. Ngo, X.T., Bhasin, S., Danger, J.-L., Guilley, S., Najm, Z.: Linear complementary dual code improvement to strengthen encoded circuit against hardware Trojan horses, pp. 82–87. IEEE International Symposium on Hardware Oriented Security and Trust (HOST), Washington (2015)

    Google Scholar 

Download references

Acknowledgements

The authors are supported by a bilateral cooperation program between Korea and Turkey. Choi and Kim are supported by NRF under the project code 2020K2A9A1A06108874. Güneri and Özbudak are supported by TÜBİTAK under the project code 120N932.

Author information

Authors and Affiliations

  1. Department of Biomedical Engineering, 50, UNIST-gil, 44919, Ulsan, Republic of Korea

    Whan-Hyuk Choi

  2. Faculty of Engineering and Natural Sciences, Sabancı University, 34956, İstanbul, Turkey

    Cem Güneri

  3. Department of Mathematics, Sogang University, 04107, Seoul, Republic of Korea

    Jon-Lark Kim

  4. Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Ferruh Özbudak

Authors
  1. Whan-Hyuk Choi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cem Güneri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jon-Lark Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ferruh Özbudak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jon-Lark Kim.

Additional information

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

Choi, WH., Güneri, C., Kim, JL. et al. Optimal Binary Linear Complementary Pairs of Codes. Cryptogr. Commun. 15, 469–486 (2023). https://doi.org/10.1007/s12095-022-00612-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-022-00612-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation