Advertisement

Bounds on Distance Distributions in Codes of Given Size

  • Gérard Cohen
  • Michael Krivelevich
  • Simon Litsyn
Chapter
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 712)

Abstract

A new upper bound on the possible distance distribution of a code of a given size is proved. The main instrument of the proof is the Beckner inequality from Harmonic Analysis. It is also shown that the obtained bound is almost tight.

Keywords

Linear Code Competitive Ratio Isoperimetric Inequality Network Security Distance Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Alon, G. Kalai, M. Ricklin, and L. Stockmeyer, Lower bounds on the competitive ratio for mobile user tracking and distributed job scheduling, Theoretical Computer Science, vol.130, 1994, pp.175–201.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    A. Ashikhmin, A. Barg, and S. Litsyn, Estimates of the distance distribution of codes and designs, IEEE Trans. Inform. Theory, vol.45, 6, 1999, pp.1808–1816.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    L. Bassalygo, G. Cohen, and G. Zémor, Codes with forbidden distances, Discrete Mathematics, 213, 2000, pp. 3–11.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    I. F. Blake, and R. C. Mullin, The Mathematical Theory of Coding, Academic Press, 1975.zbMATHGoogle Scholar
  5. [5]
    W. Beckner, Inequalities in Fourier analysis, Ann. of Math., vol.102, 1975, pp.159–182.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein, Covering Codes, Amsterdam: Elsevier, 1997.zbMATHGoogle Scholar
  7. [7]
    G. Cohen, M. Krivelevich, and S. Litsyn, manuscript.Google Scholar
  8. [8]
    G. Kalai, and N. Linial, On the distance distribution of codes, IEEE Trans. Inform. Theory, vol.41, 1995, pp.1467–1472.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    T. Klove and V. Korzhik, Error Detecting Codes, General Theory and Applications in Feedback Communication Systems, Kluwer Acad. Publ., Boston, 1995.CrossRefGoogle Scholar
  10. [10]
    N. Linial and A. Samorodnitsky, Linear codes and character sums, manuscript.Google Scholar
  11. [11]
    F. J. MacWilliams, and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, 1977.zbMATHGoogle Scholar
  12. [12]
    J. -P. Tillich, and G. Zémor, Discrete isoperimetric inequalities and the probability of a decoding error, Combin. Probab. Comput., vol. 9, 5, 2000, pp. 465–479.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    G. Zémor, and G. Cohen, The threshold probability of a code, IEEE Trans. Inform. Theory, vol.41, 2, 1995, pp. 469–477.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Gérard Cohen
    • 1
  • Michael Krivelevich
    • 2
  • Simon Litsyn
    • 3
  1. 1.Département Informatique et RéseauxENSTParisFrance
  2. 2.School of MathematicsTel Aviv UniversityTel AvivIsrael
  3. 3.Department of Electrical Engineering-SystemsTel Aviv UniversityTel AvivIsrael

Personalised recommendations