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Correctable Errors of Weight Half the Minimum Distance Plus One for the First-Order Reed-Muller Codes

  • Kenji Yasunaga
  • Toru Fujiwara
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4851)

Abstract

The number of correctable/uncorrectable errors of weight half the minimum distance plus one for the first-order Reed-Muller codes is determined. From a cryptographic viewpoint, this result immediately leads to the exact number of Boolean functions of m variables with nonlinearity 2m − 2 + 1. The notion of larger half and trial set, which is introduced by Helleseth, Kløve, and Levenshtein to describe the monotone structure of correctable/uncorrectable errors, plays a significant role in the result.

Keywords

Syndrome decoding Reed-Muller code correctable error Boolean function nonlinearity larger half 

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References

  1. 1.
    Peterson, W.W., Weldon Jr., E.J.: Error-Correcting Codes, 2nd edn. MIT Press, Cambridge (1972)zbMATHGoogle Scholar
  2. 2.
    Zémor, G.: Threshold Effects in Codes. In: Cohen, G., Lobstein, A., Zémor, G., Litsyn, S.N. (eds.) Algebraic Coding. LNCS, vol. 781, pp. 278–286. Springer, Heidelberg (1994)Google Scholar
  3. 3.
    Helleseth, T., Kløve, T., Levenshtein, V.: Error-Correction Capability of Binary Linear Codes. IEEE Trans. Infom. Theory 51(4), 1408–1423 (2005)CrossRefGoogle Scholar
  4. 4.
    Canteaut, A., Carlet, C., Charpin, P., Fontaine, C.: On Cryptographic Properties of the Cosets of R(1,m). IEEE Trans. Inform. Theory 47(4), 1513–1949 (2001)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Carlet, C.: Boolean Functions for Cryptography and Error Correcting Codes. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models, Cambridge University Press, Cambridge (press)Google Scholar
  6. 6.
    Berlekamp, E.R., Welch, L.R.: Weight Distributions of the Cosets of the (32,6) Reed-Muller Code. IEEE Trans. Inform. Theory 18(1), 203–207 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Wu, C.K.: On Distribution of Boolean Functions with Nonlinearity ≤ 2n − 2: Australasian. Journal of Combinatorics 17, 51–59 (1998)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Kenji Yasunaga
    • 1
  • Toru Fujiwara
    • 1
  1. 1.Graduate School of Information Science and Technology, Osaka University, Suita 565-0871Japan

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