Minimal codewords in Reed---Muller codes

  • Authors:

  • J. Schillewaert;L. Storme;J. A. Thas

  • Affiliations:

  • Mathematics and Statistics Department, University of Canterbury, Christchurch, New Zealand 8140;Department of Pure Mathematics and Computer Algebra, Ghent University, Ghent, Belgium 9000;Department of Pure Mathematics and Computer Algebra, Ghent University, Ghent, Belgium 9000

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2010

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Abstract

Minimal codewords were introduced by Massey (Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, pp 276---279, 1993) for cryptographical purposes. They are used in particular secret sharing schemes, to model the access structures. We study minimal codewords of weight smaller than 3 · 2 m驴r in binary Reed---Muller codes RM(r, m) and translate our problem into a geometrical one, using a classification result of Kasami and Tokura (IEEE Trans Inf Theory 16:752---759, 1970) and Kasami et al. (Inf Control 30(4):380---395, 1976) on Boolean functions. In this geometrical setting, we calculate numbers of non-minimal codewords. So we obtain the number of minimal codewords in the cases where we have information about the weight distribution of the code RM(r, m). The presented results improve previous results obtained theoretically by Borissov et al. (Discrete Appl Math 128(1), 65---74, 2003), and computer aided results of Borissov and Manev (Serdica Math J 30(2-3), 303---324, 2004). This paper is in fact an extended abstract. Full proofs can be found on the arXiv.