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Four families of minimal binary linear codes with \(w_{\min }/w_{\max }\le 1/2\)

  • Wenqin Zhang
  • Haode Yan
  • Honglei Wei
Original Paper

Abstract

As a special type of linear codes, minimal linear codes have important applications in secret sharing. Up to now, only a few infinite families of minimal binary linear codes with \(w_{\min }/w_{\max }\le 1/2\) were reported in the literature, while vast knowledge exists on the ones with \(w_{\min }/w_{\max }> 1/2\). Herein, \(w_{\min }\) and \(w_{\max }\) respectively denote the minimum and maximum nonzero Hamming weights in a linear code. Recently, several classes of linear codes with certain properties were constructed by Zhou et al. from a generic construction. The objective of this paper is to obtain four families of minimal binary linear codes with \(w_{\min }/w_{\max }\le 1/2\) from those linear codes proposed by Zhou et al. The parameters of our minimal linear codes are quite different from known ones. Based on the properties of Krawtchouk polynomials, the weight distributions of all these four families of binary linear codes are established.

Keywords

Linear codes Binary codes Minimal codes Secret sharing 

Mathematics Subject Classification

05B05 51E10 94B15 

Notes

Acknowledgements

The authors are grateful to the anonymous reviewers for careful reading and for invaluable suggestions which improve the quality of the paper. This work was supported by the National Cryptography Development Fund under Grant MMJJ20180103.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsSouthwest Jiaotong UniversityChengduChina
  2. 2.State Key Laboratory of CryptologyBeijingChina
  3. 3.School of Economics and Management, National-Local Joint Engineering Laboratory of System Credibility Automatic VerificationSouthwest Jiaotong UniversityChengduChina

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