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Further projective binary linear codes derived from two-to-one functions and their duals

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Abstract

Binary linear codes with few weights have wide applications in communication, secret sharing schemes, authentication codes, association schemes, strongly regular graphs, etc. Projective binary linear codes are among the most important subclasses of binary linear codes for practical applications. In this paper, motivated by the two excellent recent papers (Li et al. in IEEE Trans Inf Theory 67(7):4263–4275, 2021) and (Wang et al. in IEEE Trans Inf Theory 67(8):5133–5148, 2021), several new families of few-weight projective binary linear codes are constructed from the defining sets, and then their Hamming weight distributions are determined by employing the Walsh transform of the corresponding two-to-one functions over finite fields with the even characteristic. Our constructions can produce binary linear codes with new parameters. Some of the constructed binary linear codes are optimal or almost optimal according to the online Database of Grassl, and the duals of some of them are distance-optimal with respect to the sphere packing bound. This paper also shows once again that the two-to-one functions initially studied in (Mesnager and Qu in IEEE Trans Inf Theory 65(12):7884–7895, 2019) are also promising objects in coding theory. Although our derived codes use objects considered in the very recent literature, the analysis of our designed codes involves functions having different algebraic structures (and, therefore, other Walsh transform distribution) and requires solving new systems of equations over finite fields, which is an essential step in determining the weight distribution of our constructed codes. As applications, some of the codes presented in this paper can be used to construct association schemes and secret sharing schemes with interesting access structures.

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Acknowledgements

The authors would like to express their thanks and gratitude to the anonymous reviewers for their valuable comments and helpful suggestions, which have greatly improved the quality of this paper. This work was done while L. Qian was visiting S. Mesnager in the department of Mathematics of the University of Paris VIII, at the Laboratory LAGA, France, in cooperation with X. Cao. It has been funded by China Scholarship Council, National Natural Science Foundation of China under the Grant numbers 11771007, 12171241, 62172183, and Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant number KYCX21_0175. S. Mesnager is supported by the French Agence Nationale de la Recherche through ANR BARRACUDA (ANR-21-CE39-0009).

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Authors and Affiliations

  1. Department of Mathematics, University of Paris VIII, 93526, Saint-Denis, France

    Sihem Mesnager & Liqin Qian

  2. University Sorbonne Paris Cité, LAGA, UMR 7539, CNRS, 93430, Villetaneuse, France

    Sihem Mesnager

  3. Telecom Paris, Polytechnic Institute of Paris, 91120, Palaiseau, France

    Sihem Mesnager

  4. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, Jiangsu, China

    Liqin Qian & Xiwang Cao

  5. Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 211106, China

    Xiwang Cao

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  1. Sihem Mesnager
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  2. Liqin Qian
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  3. Xiwang Cao
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Correspondence to Liqin Qian.

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Communicated by L. Storme.

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Appendix

Appendix

The Appendix provides the Hamming weights and their weight distributions for the codes presented in Sect. 3.

where \(A_{w_2}+A_{w_4}=2^{3m-3}\) and \(A_{w_1}+A_{w_5}=2^{3m-5}-2^{2m-4}\).

where \(A_{w_2}+A_{w_4}=2^{3m-2}\) and \(A_{w_1}+A_{w_5}=2^{3m-4}-2^{2m-3}\).

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Mesnager, S., Qian, L. & Cao, X. Further projective binary linear codes derived from two-to-one functions and their duals. Des. Codes Cryptogr. 91, 719–746 (2023). https://doi.org/10.1007/s10623-022-01122-3

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  • DOI: https://doi.org/10.1007/s10623-022-01122-3

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