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A new method for constructing linear codes with small hulls

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Abstract

The hull of a linear code over finite fields is the intersection of the code and its dual, which was introduced by Assmus and Key. In this paper, we develop a method to construct linear codes with trivial hull (LCD codes) and one-dimensional hull by employing the positive characteristic analogues of Gauss sums. These codes are quasi-abelian, and sometimes doubly circulant. Some sufficient conditions for a linear code to be an LCD code (resp. a linear code with one-dimensional hull) are presented. It is worth mentioning that we present a lower bound on the minimum distances of the constructed linear codes. As an application, using these conditions, we obtain some optimal or almost optimal LCD codes (resp. linear codes with one-dimensional hull) with respect to the online Database of Grassl.

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Acknowledgements

The authors deeply thank the editor and the anonymous reviewers for their valuable comments which have highly improved the quality of the paper. This research is supported by National Natural Science Foundation of China under Grant 11771007, 12171241, 62172183 and Postgraduate Research and Practice Innovation Program of Jiangsu Province under Grant KYCX21_0175 and China Scholarship Council.

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Authors and Affiliations

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

    Liqin Qian & Xiwang Cao

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

    Xiwang Cao

  3. School of Mathematics, Southeast University, Nanjing, 211189, Jiangsu, China

    Wei Lu

  4. I2M(CNRS, Aix-Marseille University, Centrale Marseille), Marseille, France

    Patrick Solé

Authors
  1. Liqin Qian

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  2. Xiwang Cao

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  3. Wei Lu

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  4. Patrick Solé

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Correspondence toXiwang Cao.

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This is one of several papers published inDesigns, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”

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Qian, L., Cao, X., Lu, W.et al. A new method for constructing linear codes with small hulls.Des. Codes Cryptogr.90, 2663–2682 (2022). https://doi.org/10.1007/s10623-021-00940-1

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Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless

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