Characterization and classification of optimal LCD codes

Designs, Codes and Cryptography (2021)Cite this article

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 [n, k] 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 [n, k] 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/.

Download references

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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Affiliations

  1. Department of Computer Science, Shizuoka University, Hamamatsu, 432–8011, Japan

    Makoto Araya

  2. Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai, 980–8579, Japan

    Masaaki Harada & Ken Saito

Authors
  1. Makoto Araya
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  2. Masaaki Harada
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  3. Ken Saito
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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