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)


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.


Cyclic Code Arithmetic Code Galois Ring Improve Bound Homogeneous Weight 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [An]
    Anderson, D.R.: A new class of cyclic codes. SIAM J. Appl. Math. 16, 181–197 (1968)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle Scholar
  4. [He]
    Helleseth, T.: On the covering radius of cyclic linear codes and arithmetic codes. Disc. Appl. Math. 11, 157–173 (1985)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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