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Complete weight enumerators of few-weight linear codes

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Abstract

Few-weight linear codes have important applications in the construction of strongly regular graphs, authentication codes and secret sharing schemes. In this paper, some few-weight linear codes are constructed from proper defining sets over finite fields. Their complete weight enumerators are explicitly determined using Weil sums. As applications, we give two classes of new projective three-weight linear codes, which achieve the Griesmer bound. We construct some new strongly regular graphs and infinite families of minimal three-weight linear codes with \(\frac{w_{min}}{w_{max}}\le \frac{p-1}{p}\). Moreover, some new authentication codes are presented. Our results generalize and improve the work of Zhu and Liao (Finite Fields Appl. 75, 101897, 2021).

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Acknowledgements

Jian Gao is supported by the Shandong Provincial Natural Science Foundation (Grant Nos. ZR2024YQ057, ZR2022MA024), the National Natural Science Foundation of China (Grant Nos. 12071264, 11701336) and the IC Program of Shandong Institutions of Higher Learning For Youth Innovative Talents. Fang-Wei Fu is supported by the National Key Research and Development Program of China (Grant No. 2018YFA0704703), the National Natural Science Foundation of China (Grant No. 61971243), the Natural Science Foundation of Tianjin (20JCZDJC00610), the Fundamental Research Funds for the Central Universities of China (Nankai University). The authors would like to thank the anonymous reviewers and the Editor Prof. Claude Carlet for their valuable suggestions and comments that helped to greatly improve the article.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, China

    Xiangdi Zeng & Jian Gao

  2. Chern Institute of Mathematics and LPMC, Tianjin Key Laboratory of Network and Data Security Technology, Nankai University, Tianjin, 300071, China

    Xiangrui Meng & Fang-Wei Fu

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  1. Xiangdi Zeng
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  2. Xiangrui Meng
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  3. Jian Gao
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  4. Fang-Wei Fu
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Jian Gao introduced the problem and gave the idea to solve it. Xiangdi Zeng and Xiangrui Meng solved the problem and wrote the manuscript. All authors reviewed the manuscript.

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Correspondence to Jian Gao.

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Zeng, X., Meng, X., Gao, J. et al. Complete weight enumerators of few-weight linear codes. Cryptogr. Commun. 17, 1013–1050 (2025). https://doi.org/10.1007/s12095-025-00804-8

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