Advertisement

A transformation on boolean functions, its consequences on some problems related to reed-muller codes

  • Claude Carlet
Section 1 Algebraic Codes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 514)

Abstract

We introduce a transformation defined on the set of all boolean functions defined on Galois fields GF(2m), m∈ℕ*,(or on F 2 m ), which changes their weights in a way easy to be followed, and which, when we iterate it, reduces their degrees down to 2 or 3. We deduce that it is as difficult to find a general characterization of the weights in the Reed-Muller codes of order 3 as it is to obtain one in the Reed-Muller codes of any orders. We also use this transformation to characterize the existence of some affine sets of bent functions, and to obtain bent functions of degree 4 from bent functions of degree 3.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. (1).
    C. Carlet, thèse, publication du LITP, Institut Blaise Pascal, Université Paris 6, no 90.59 (1990).Google Scholar
  2. (2).
    J. F. Dillon, "Elementary Hadamard Difference sets", Ph. D. Thesis, Univ. of Mariland (1984).Google Scholar
  3. (3).
    T. Kasami and N. Tokura "On the weight structure of the Reed Muller codes", IEEE Trans. Info. Theory 16 (1970) 752–759.CrossRefGoogle Scholar
  4. (4).
    F. J. Mac Williams and N. J. Sloane, "The theory of error-correcting codes", Amsterdam, North Holland.Google Scholar
  5. (5).
    O. S. Rothaus, "On bent functions", J. Comb. Theory, 20A (1976) 300–305CrossRefGoogle Scholar
  6. (6).
    J. H. Van Lint, "Coding Theory", Springer Verlag 201.Google Scholar
  7. (7).
    H.C.A. Van Tilborg "On weights in codes" Report 71-WSK03, Dep.of math. Technological University of Eindhoven, Netherlands (1971).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Claude Carlet
    • 1
    • 2
  1. 1.Université de PicardieFrance
  2. 2.INRIA, Bat 10, Domaine de VoluceauLe Chesnay, CedexFrance

Personalised recommendations