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Linear codes with one-dimensional hull associated with Gaussian sums

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Abstract

The hull of a linear code over finite fields, the intersection of the code and its dual, has been of interest and extensively studied due to its wide applications. For example, it plays a vital role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and for computing the automorphism group of a linear code. People are interested in pursuing linear codes with small hulls since, for such codes, the aforementioned algorithms are very efficient. In this field, Carlet, Mesnager, Tang and Qi gave a systematic characterization of LCD codes, i.e, linear codes with null hull. In 2019, Carlet, Li and Mesnager presented some constructions of linear codes with small hulls. In the same year, Li and Zeng derived some constructions of linear codes with one-dimensional hull by using some specific Gaussian sums. In this paper, we use general Gaussian sums to construct linear codes with one-dimensional hull by utilizing number fields, which generalizes some results of Li and Zeng (IEEE Trans. Inf. Theory 65(3), 1668–1676, 2019) and also of those presented by Carlet et al. (Des. Codes Cryptogr. 87(12), 3063–3075, 2019). We give sufficient conditions to obtain such codes. Notably, some codes we obtained are optimal or almost optimal according to the Database. This is the first attempt on constructing linear codes by general Gaussian sums which have one-dimensional hull and are optimal. Moreover, we also develop a bound of on the minimum distances of linear codes we constructed.

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Acknowledgments

The authors deeply thank the Assoc. Edit. and the anonymous reviewers for their valuable comments which have highly improved the quality of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211100, China

    Liqin Qian & Xiwang Cao

  2. Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100042, China

    Xiwang Cao

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

    Sihem Mesnager

  4. University of Paris XIII, CNRS, LAGA UMR 7539, Sorbonne Paris Cité, 93430, Villetaneuse, France

    Sihem Mesnager

  5. Telecom Paris, 75013, Paris, France

    Sihem Mesnager

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

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This research is supported by the National Natural Science Foundation of China under Grant 11771007 and Grant 61572027.

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Qian, L., Cao, X. & Mesnager, S. Linear codes with one-dimensional hull associated with Gaussian sums. Cryptogr. Commun. 13, 225–243 (2021). https://doi.org/10.1007/s12095-020-00462-y

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