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Several constructions of optimal LCD codes over small finite fields

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Abstract

Linear complementary dual (LCD) codes are linear codes which intersect their dual codes trivially, which have been of interest and extensively studied due to their practical applications in computational complexity and information protection. In this paper, we give some methods for constructing LCD codes over small finite fields by modifying some typical methods for constructing linear codes. We show that all odd-like binary Euclidean LCD codes, ternary Euclidean LCD codes and quaternary Hermitian LCD codes can be constructed using the modified methods. Our results improve the known lower bounds on the largest minimum distances of LCD codes. Furthermore, we give two counterexamples to disprove the conjecture proposed by Bouyuklieva (Des. Codes Cryptogr. 89(11), 2445–2461 2021).

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Availability of supporting data

The data that support the findings of this study are openly available at https://ahu-coding.github.io/code1/.

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Acknowledgements

The authors would like to thank Dr. Hongwei Zhu for helpful discussions. The authors would also like to thank the National Natural Science Foundation of China (12071001) for funding this research. The authors would also like to thank the editor and the anonymous referees for helpful comments which have highly improved the quality of the paper.

Funding

This research is supported by the National Natural Science Foundation of China (12071001).

Author information

Authors and Affiliations

  1. Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education, School of Mathematical Sciences, Anhui University, 230601, Hefei, China

    Shitao Li & Minjia Shi

  2. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, 100093, Beijing, China

    Shitao Li & Minjia Shi

  3. State Grid Anhui Electric Power Co., Ltd, Hefei, China

    Huizhou Liu

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  1. Shitao Li
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  2. Minjia Shi
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  3. Huizhou Liu
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All authors reviewed the manuscript.

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Correspondence to Minjia Shi.

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Appendix

Appendix

Table 1 Binary Euclidean LCD codes from Theorem 3.2
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Table 2 Binary Euclidean LCD codes from Theorems 3.5 and 3.10
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Table 3 The binary LCD codes from some inequalities
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Table 4 Bounds on the minimum diatance of binary Euclidean LCD codes
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Table 5 Bounds on the minimum diatance of binary Euclidean LCD codes
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Table 6 Ternary Euclidean LCD codes from Theorems 3.2 and 3.3
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Table 7 Ternary Euclidean LCD codes from Theorems 3.5 and 3.10
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Table 8 Bounds on the minimum distances of ternary Euclidean LCD codes
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Table 9 Bounds on the minimum distances of ternary Euclidean LCD codes
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Li, S., Shi, M. & Liu, H. Several constructions of optimal LCD codes over small finite fields. Cryptogr. Commun. (2024). https://doi.org/10.1007/s12095-024-00699-x

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