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Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 463–480 | Cite as

Linear codes from weakly regular plateaued functions and their secret sharing schemes

  • Sihem Mesnager
  • Ferruh Özbudak
  • Ahmet SınakEmail author
Article
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

Abstract

Linear codes, the most significant class of codes in coding theory, have diverse applications in secret sharing schemes, authentication codes, communication, data storage devices and consumer electronics. The main objectives of this paper are twofold: to construct three-weight linear codes from plateaued functions over finite fields, and to analyze the constructed linear codes for secret sharing schemes. To do this, we generalize the recent contribution of Mesnager given in (Cryptogr Commun 9(1):71–84, 2017). We first introduce the notion of (non)-weakly regular plateaued functions over \({{\mathbb {F}}}_p\), with p being an odd prime. We next construct three-weight linear p-ary (resp. binary) codes from weakly regular p-ary plateaued (resp. Boolean plateaued) functions and determine their weight distributions. We finally observe that the constructed linear codes are minimal for almost all cases, which implies that they can be directly used to construct secret sharing schemes with nice access structures. To the best of our knowledge, the construction of linear codes from plateaued functions over \({{\mathbb {F}}}_p\), with p being an odd prime, is studied in this paper for the first time in the literature.

Keywords

Binary linear codes Linear p-ary codes Secret sharing schemes Weakly regular plateaued functions Weight distribution 

Mathematics Subject Classification

94A60 14G50 11T71 

Notes

Acknowledgements

The authors extend thanks to the Editor and anonymous reviewers for their valuable comments and suggestions, which improved the quality and presentation of the manuscript. The third author is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK), Program No: BİDEB 2214/A.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics at University of Paris VIII, LAGA, UMR 7539, CNRS at University of Paris XIII and Telecom ParisTechParisFrance
  2. 2.Department of Mathematics, Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  3. 3.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  4. 4.Department of Mathematics and Computer SciencesNecmettin Erbakan UniversityKonyaTurkey
  5. 5.LAGA, UMR 7539, CNRS at Universities of Paris VIII and XIIIParisFrance

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