Characterization and classification of optimal LCD codes

Abstract

Linear complementary dual (LCD) codes are linear codes that intersect with their dual trivially. We give a characterization of LCD codes over \(\mathbb {F}_q\) having large minimum weights for \(q \in \{2,3\}\). Using the characterization, for arbitrary n, we determine the largest minimum weights among LCD [nk] codes over \(\mathbb {F}_q\), where \((q,k) \in \{(2,4), (3,2),(3,3)\}\). Moreover, for arbitrary n, we give a complete classification of optimal LCD [nk] codes over \(\mathbb {F}_q\), where \((q,k) \in \{(2,3), (2,4), (3,2),(3,3)\}\).

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References

  1. 1.

    Araya M., Harada M.: On the classification of linear complementary dual codes. Discret. Math. 342, 270–278 (2019).

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bonisoli A.: Every equidistant linear code is a sequence of dual Hamming codes. Ars Combin. 18, 181–186 (1984).

    MathSciNet  MATH  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cameron P.J., van Lint J.H.: Designs, Codes and Their Links. Cambridge University Press, Cambridge (1991).

    Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 8.

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

    MathSciNet  Article  Google Scholar 

  9. 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. 10.

    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, 1237–1245 (2019).

    MathSciNet  Article  Google Scholar 

  11. 11.

    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).

    MathSciNet  Article  Google Scholar 

  12. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

    Harada M., Saito K.: Remark on subcodes of linear complementary dual codes. Inf. Process. Lett. 159, 10596 (2020).

    MathSciNet  MATH  Google Scholar 

  14. 14.

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

    Google Scholar 

  15. 15.

    Lina Jr. E.R., Nocon E.G.: On the construction of some LCD codes over finite fields. Manila J. Sci. 9, 67–82 (2016).

    Google Scholar 

  16. 16.

    Lu L., Li R., Guo L., Fu Q.: Maximal entanglement entanglement-assisted quantum codes constructed from linear codes. Quant. Inf. Process. 14, 165–182 (2015).

    MathSciNet  Article  Google Scholar 

  17. 17.

    Maruta T.: On the achievement of the Griesmer bound. Des. Codes Cryptogr. 12, 83–87 (1997).

    MathSciNet  Article  Google Scholar 

  18. 18.

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

  19. 19.

    McKay B.D., Piperno A.: Practical graph isomorphism, II. J. Symb. Comput. 60, 94–112 (2014).

    MathSciNet  Article  Google Scholar 

  20. 20.

    Pang B., Zhu S., Kai X.: Some new bounds on LCD codes over finite fields. Cryptogr. Commun. 12, 743–755 (2020).

    MathSciNet  Article  Google Scholar 

  21. 21.

    Shoup V. NTL: A Library for doing Number Theory. http://www.shoup.net/ntl/.

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Acknowledgements

The authors would like to thank Tatsuya Maruta for his useful comments. This work was supported by JSPS KAKENHI Grant Number 19H01802.

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Correspondence to Masaaki Harada.

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Araya, M., Harada, M. & Saito, K. Characterization and classification of optimal LCD codes. Des. Codes Cryptogr. (2021). https://doi.org/10.1007/s10623-020-00834-8

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Keywords

  • Linear complementary dual code
  • Binary code
  • Ternary code
  • Simple code
  • Griesmer bound

Mathematics Subject Classification

  •   94B05