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Weight distributions of Q2DC codes over finite fields

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Abstract

Quasi-2-dimensional cyclic (Q2DC) codes over finite fields are important linear codes. They are asymptotically good, and have deep connection with convolutional codes. In this paper, using the trace representation and Gauss sums, we determine the Hamming weight distribution of Q2DC codes. As an application, we construct some classes of constant weight Q2DC codes achieving the Griesmer bound and some classes of two-weight, three-weight Q2DC codes. Moreover, by two-weight Q2DC codes, we obtain secret sharing schemes and association schemes under some special cases.

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Acknowledgements

Jian Gao is supported by the Shandong Provincial Natural Science Foundation (No. 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). Fanghui Ma is supported by the Shandong Provincial Natural Science Foundation (No.ZR2021QA047).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, People’s Republic of China

    Xiangrui Meng, Jian Gao & Fanghui Ma

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

    Fang-Wei Fu

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  1. Xiangrui Meng
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Correspondence to Jian Gao.

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Communicated by D. Panario.

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Meng, X., Gao, J., Fu, FW. et al. Weight distributions of Q2DC codes over finite fields. Des. Codes Cryptogr. 91, 807–830 (2023). https://doi.org/10.1007/s10623-022-01128-x

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  • DOI: https://doi.org/10.1007/s10623-022-01128-x

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