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Several classes of new weakly regular bent functions outside \(\mathcal{R}\mathcal{F}\), their duals and some related (minimal) codes with few weights

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Abstract

Boosted by cryptography and coding theory applications and rich connections to objects from geometry and combinatorics, bent functions and related functions developed into a lively research area. In the mid-seventies, Rothaus initially introduced bent functions in the Boolean case, but later, they extended in the p-ary case where p is any prime. Such an extension brought more rich connections to bent functions. In addition to the theory of Fourier transform, handling p-ary bent functions (where p is an odd prime), analyzing their properties and designing them require using the theory of cyclotomic fields. Such a family is classified into weakly regular bent and regular bent. Compared to the Boolean case, the class of p-ary bent functions inherits a much larger variety of properties. For example, it contains the class of dual-bent functions as a proper subclass, which again includes weakly regular bent functions as a proper subclass. Weakly regular bent functions can also be employed in many domains. In particular, they have been widely used in designing good linear codes for several applications (such as secret sharing and two-party computation), association schemes, and strongly regular graphs. More exploration is still needed despite a lot of interest and attention in this topic to increase our knowledge of bent functions and design them. In particular, only a few constructions of weakly regular bent functions have been presented in the literature. Nice construction methods of these functions have been given by Tang et al (IEEE Trans Inf Theory 62(3):1166–1176, 2016). All the known infinite families of weakly regular bent functions belong to a class (denoted by Tang et al.  in the previous reference) \(\mathcal{R}\mathcal{F}\). This paper is devoted to weakly regular bent functions, and its objective is twofold. First, it aims to generate new infinite families of weakly regular bent functions living outside \(\mathcal{R}\mathcal{F}\) and secondly, to exploit the constructed functions to design new families of p-ary linear codes and investigate their use for some standard application after studying this minimality based on their weight distributions. More specifically, we present several classes of weakly regular bent functions obtained from monomial bent functions by modifying the values of some known weakly regular bent functions on some subsets of the finite field \(\mathbb {F}_{p^n}\) (n is a positive integer). Our weakly regular bent functions are of degree either p or \(\frac{p+1}{2}\). We also explicitly determine their corresponding dual functions. Finally, we exploit our constructions to derive four new classes of linear codes with three to seven weights. We also show that two of them lead to minimal codes, which are more appropriate for several concrete applications.

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Acknowledgements

The authors address their thanks to the Associate Editor and the reviewers for their valuable comments and constructive suggestions that improved the quality of this paper highly. Xiaoni Du was partially supported by the National Natural Science Foundation of China under Grant no. 62172337 and Guangxi Key Laboratory of Cryptography and Information Security (No. GCI201910). Sihem Mesnager was supported by the French Agence Nationale de la Recherche through ANR BARRACUDA (ANR-21-CE39-0009).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, Gansu, People’s Republic of China

    Xiaoni Du & Wengang Jin

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

    Sihem Mesnager

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

    Sihem Mesnager

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

    Sihem Mesnager

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  1. Xiaoni Du
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Correspondence to Sihem Mesnager.

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Du, X., Jin, W. & Mesnager, S. Several classes of new weakly regular bent functions outside \(\mathcal{R}\mathcal{F}\), their duals and some related (minimal) codes with few weights. Des. Codes Cryptogr. 91, 2273–2307 (2023). https://doi.org/10.1007/s10623-023-01198-5

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