Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the coefficients of binary bent functions


Author: Xiang-dong Hou
Journal: Proc. Amer. Math. Soc. 128 (2000), 987-996
MSC (1991): Primary 05B10, 94B27; Secondary 94A60
DOI: https://doi.org/10.1090/S0002-9939-99-05146-1
Published electronically: August 17, 1999
MathSciNet review: 1641634
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a 2-adic inequality for the coefficients of binary bent functions in their polynomial representations. The 2-adic inequality implies a family of identities satisfied by the coefficients. The identities also lead to the discovery of some new affine invariants of Boolean functions on ${\mathbf Z}_2^m$.


References [Enhancements On Off] (What's this?)

  • 1. E. Berlekamp and L. R. Welch, Weight distribution of the cosets of the (32,6) Reed-Muller code, IEEE Trans. Inform. Theory 18 (1972), 203-207. MR 52:16844
  • 2. C. Carlet, Two new classes of bent functions, Lecture Notes in Computer Science 765, Springer-Verlag, Berlin, 1994, 77-101. MR 95f:94016
  • 3. C. Carlet, Generalized partial spreads, IEEE Trans. Inform. Theory 41 (1995), 1482-1487. MR 97b:94043
  • 4. C. Carlet, A construction of bent functions, London Math. Soc. Lecture Series 233, Cambridge Univ. Press, 1996, 47-58. MR 97k:94081
  • 5. C. Carlet and P. Guillot, A characterization of binary bent functions, J. Combin. Theory Ser A 76 (1996), 328-335. MR 99b:94054
  • 6. H. Chung and P. V. Kummar, A new general construction for generalized bent functions, IEEE Trans. Inform. Theory 35 (1989), 206-209. CMP 21:12
  • 7. J. F. Dillon, Elementary Hadamard Difference Sets, Ph.D. Thsis, Univ. of Maryland, 1974.
  • 8. H. Dobbertin, Constructions of bent functions and balanced Boolean functions with high nonlinearity, Lecture Notes in Computer Science 1008, Springer-Verlag, Berlin, 1995, 61-74.
  • 9. X. Hou, $AGL(m,2)$ acting on $R(r,m)/R(s,m)$, J. Algebra 171 (1995), 921-938. MR 96j:94023
  • 10. X. Hou, $GL(m,2)$ acting on $R(r,m)/R(r-1,m)$, Discrete Math. 149 (1996), 99-122. MR 97g:94030
  • 11. X. Hou, $q$-ary bent functions constructed from chain rings, Finite Fields Appl. 4 (1998), 55-61. CMP 98:09
  • 12. X. Hou, Cubic bent functions, Discrete Math. 189 (1998), 149-161.
  • 13. X. Hou, New constructions of bent functions, J. Statistical Planning and Inference, to appear.
  • 14. X. Hou and P. Langevin, Results on bent functions, J. Combin. Theory Ser A 80 (1997), 232-246. MR 99b:05025
  • 15. P. V. Kumar, R. A. Scholtz and L. R. Welch, Generalized bent functions and their properties, J. Combin. Theory Ser A 40 (1985), 90-107. MR 87i:05075
  • 16. P. Langevin, On generalized bent functions, CISM Courses and Lectures 339, Springer-Verlag, Wien, 1992, 147-157. MR 95d:11166
  • 17. J. Maiorana, A classification of the cosets of the Reed-Muller code $R(1,6)$, Math. Comp. 57 (1991), 403-414. MR 91j:94023
  • 18. R. L. McFarland, A family of difference sets in noncyclic groups, J. Combin. Theory Ser A 15 (1973), 1-10. MR 47:3198
  • 19. O. S. Rothaus, On ``bent'' functions, J. Combin. Theory Ser A 20 (1976), 300-305. MR 53:7797

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 05B10, 94B27, 94A60

Retrieve articles in all journals with MSC (1991): 05B10, 94B27, 94A60


Additional Information

Xiang-dong Hou
Affiliation: Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435
Email: xhou@euler.math.wright.edu

DOI: https://doi.org/10.1090/S0002-9939-99-05146-1
Keywords: Bent function, Boolean function, affine invariant
Received by editor(s): January 12, 1998
Received by editor(s) in revised form: June 12, 1998
Published electronically: August 17, 1999
Additional Notes: This work was supported by a grant from the Research Council of Wright State University.
Communicated by: John R. Stembridge
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society