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The linear codes of t-designs held in the Reed-Muller and Simplex codes

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Abstract

A fascinating topic of combinatorics is the study of t-designs, which has a very long history. The incidence matrix of a t-design generates a linear code over GF(q) for any prime power q, which is called the linear code of the t-design over GF(q). On the other hand, some linear codes hold t-designs with t ≥ 1. The purpose of this paper is to study the linear codes of t-designs held in the Reed-Muller and Simplex codes. Some general theory for the linear codes of t-designs held in linear codes is presented. Several open problems are also presented.

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Acknowledgments

The authors are grateful to the reviewers for their comments that improved the presentation of this paper.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

    Cunsheng Ding

  2. The School of Mathematics and Information, China West Normal University, Nanchong, 637002, China

    Chunming Tang

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  1. Cunsheng Ding
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  2. Chunming Tang
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Correspondence to Chunming Tang.

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This article belongs to the Topical Collection: Boolean Functions and Their Applications V

Guest Editors: Lilya Budaghyan, Claude Carlet, Tor Helleseth and Kaisa Nyberg

C. Ding’s research was supported by the Hong Kong Research Grants Council, Proj. No. 16300418. C. Tang was supported by National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (14E013, CXTD2014-4 and the Meritocracy Research Funds).

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Ding, C., Tang, C. The linear codes of t-designs held in the Reed-Muller and Simplex codes. Cryptogr. Commun. 13, 927–949 (2021). https://doi.org/10.1007/s12095-021-00470-6

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  • DOI: https://doi.org/10.1007/s12095-021-00470-6

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