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Hulls of linear codes from simplex codes

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Abstract

The hull of a linear code plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. Regarding the quantum error correction, linear codes with determined hull are used to construct quantum codes. In this paper, we focus on the hull of Simplex codes and punctured Simplex codes. We firstly study the properties of the matrix produced by the column vectors of a projective space and determine the Euclidean and Hermitian hull of punctured Simplex codes completely. Secondly, we investigate the Euclidean and Hermitian hull of several classes of linear codes from Simplex codes.

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Acknowledgements

G. Xu is supported by the National Natural Science Foundation of China under Grants 62172183, 12371339, the PhD Research Startup Fund for Anhui Jianzhu University (No. 2023QDZ29), the innovation team of operation research and combinatorial optimization of Anhui province (No. 2023AH010020), the Natural Science Research Foundation of Anhui Provincial Department of Education (No. 2023AH050194) and the Projects of Natural Science Research in Anhui Colleges and Universities (No. HYB20230132). G. Luo and X. Cao are supported by the National Natural Science Foundation of China under Grant 12171241 and the Natural Science Foundation of Jiangsu Province under Grant BK20230867. H. Xu is supported by the National Natural Science Foundation of China under Grant 12171134 and the Natural Science Foundation for the Higher Education Institutions of Anhui Province of China under Grant KJ2021A0926.

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Authors and Affiliations

  1. School of Mathematics and Physics, Anhui Jianzhu University, Hefei, 230601, China

    Guangkui Xu

  2. School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore

    Gaojun Luo

  3. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Xiwang Cao

  4. Key Laboratory of Mathematical Modeling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 210016, China

    Xiwang Cao

  5. School of Mathematics and Statistics, Hefei Normal University, Hefei, 230601, China

    Heqian Xu

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  1. Guangkui Xu
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  2. Gaojun Luo
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  3. Xiwang Cao
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  4. Heqian Xu
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Correspondence to Gaojun Luo.

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Communicated by Y. Zhou.

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Xu, G., Luo, G., Cao, X. et al. Hulls of linear codes from simplex codes. Des. Codes Cryptogr. 92, 1095–1112 (2024). https://doi.org/10.1007/s10623-023-01331-4

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