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Minimal codewords in Reed-Muller codes

Jeroen Schillewaert (UGent) , Leo Storme (UGent) and Joseph Thas (UGent)
(2010) DESIGNS CODES AND CRYPTOGRAPHY. 54(3). p.273-286
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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 center dot 2 (m-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.
Keywords
Reed-Muller codes, Minimal codewords, Quadrics, Affine spaces

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Citation

Please use this url to cite or link to this publication:

Chicago
Schillewaert, Jeroen, Leo Storme, and Joseph Thas. 2010. “Minimal Codewords in Reed-Muller Codes.” Designs Codes and Cryptography 54 (3): 273–286.
APA
Schillewaert, Jeroen, Storme, L., & Thas, J. (2010). Minimal codewords in Reed-Muller codes. DESIGNS CODES AND CRYPTOGRAPHY, 54(3), 273–286.
Vancouver
1.
Schillewaert J, Storme L, Thas J. Minimal codewords in Reed-Muller codes. DESIGNS CODES AND CRYPTOGRAPHY. 2010;54(3):273–86.
MLA
Schillewaert, Jeroen, Leo Storme, and Joseph Thas. “Minimal Codewords in Reed-Muller Codes.” DESIGNS CODES AND CRYPTOGRAPHY 54.3 (2010): 273–286. Print.
@article{978956,
  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 center dot 2 (m-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.},
  author       = {Schillewaert, Jeroen and Storme, Leo and Thas, Joseph},
  issn         = {0925-1022},
  journal      = {DESIGNS CODES AND CRYPTOGRAPHY},
  language     = {eng},
  number       = {3},
  pages        = {273--286},
  title        = {Minimal codewords in Reed-Muller codes},
  url          = {http://dx.doi.org/10.1007/s10623-009-9323-x},
  volume       = {54},
  year         = {2010},
}

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