Advertisement

Improved Bounds on Weil Sums over Galois Rings and Homogeneous Weights

  • San Ling
  • Ferruh Özbudak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3969)

Abstract

We generalize a recent improvement for the bounds of Weil sums over Galois rings of characteristic p 2 to Galois rings of any characteristic p l . Our generalization is not as strong as for the case p 2 and we indicate the reason. We give a class of homogeneous weights, including the homogeneous weight defined by Constantinescu and Heise, and we show their relations. We also give an application of our improvements on the homogeneous weights of some codewords.

Keywords

Cyclic Code Arithmetic Code Galois Ring Improve Bound Homogeneous Weight 
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. [An]
    Anderson, D.R.: A new class of cyclic codes. SIAM J. Appl. Math. 16, 181–197 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [B-M]
    Blake, I.F., Mark, J.W.: A note on complex sequences with low correlations. IEEE Trans. Inform. Theory 28(5), 814–816 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [C-H]
    Constantinescu, I., Heise, T.: A metric for codes over residue class rings of integers. Problemy Peredachi Informatsii 33(3), 22–28 (1997)MathSciNetzbMATHGoogle Scholar
  4. [He]
    Helleseth, T.: On the covering radius of cyclic linear codes and arithmetic codes. Disc. Appl. Math. 11, 157–173 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [H-K-M-S]
    Helleseth, T., Kumar, P.V., Moreno, O., Shanbhag, A.G.: Improved estimates via exponential sums for the minimum distance of ℤ4-linear trace codes. IEEE Trans. Inform. Theory 42(4), 1212–1216 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [K-H-C]
    Kumar, P.V., Helleseth, T., Calderbank, A.R.: An upper bound for Weil exponential sums over Galois rings with applications. IEEE Trans. Inform. Theory 41(2), 456–468 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [L-O]
    Ling, S., Özbudak, F.: An improvement on the bounds of Weil exponential sums over Galois rings with some applications. IEEE Trans. Inform. Theory 50(10), 2529–2539 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [L-O2]
    Ling, S., Özbudak, F.: Improved p-ary codes and sequence families from Galois rings of characteristic p 2. SIAM J. Discrete Math. 19(4), 1011–1028 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Lu]
    Lucas, E.: Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques suivant un module premier. Bull. Soc. Math. France 6, 49–54 (1878)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Si]
    Sidelnikov, V.M.: On mutual correlation of sequences. Soviet Math. Dokl. 12(1), 197–201 (1971)MathSciNetGoogle Scholar
  11. [T]
    Tietäväinen, A.: On the covering radius of long BCH codes. Disc. Appl. Math. 16, 75–77 (1987)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • San Ling
    • 1
  • Ferruh Özbudak
    • 2
  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeRepublic of Singapore
  2. 2.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey

Personalised recommendations