Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

INFINITE FAMILIES OF RECURSIVE FORMULAS GENERATING POWER MOMENTS OF TERNARY KLOOSTERMAN SUMS WITH SQUARE ARGUMENTS ASSOCIATED WITH O-(2n, q)

  • Kim, Dae-San
  • Received : 2009.09.11
  • Published : 2011.03.01

Abstract

In this paper, we construct eight infinite families of ternary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group $SO^-$(2n, q). Here q is a power of three. Then we obtain four infinite families of recursive formulas for power moments of Kloosterman sums with square arguments and four infinite families of recursive formulas for even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of "Gauss sums" for the orthogonal groups $O^-$(2n, q).

Keywords

index terms-Kloosterman sum;orthogonal group;special orthogonal group;double cosets;maximal parabolic subgroup;Pless power moment identity;weight distribution

File

Download PDF

References

  1. R. J. Evans, Seventh power moments of Kloosterman sums, Israel J. Math. 175 (2010), 349-362. https://doi.org/10.1007/s11856-010-0014-0
  2. G. van der Geer, R. Schoof, and M. van der Vlugt, Weight formulas for ternary Melas codes, Math. Comp. 58 (1992), no. 198, 781-792. https://doi.org/10.1090/S0025-5718-1992-1122080-4
  3. K. Hulek, J. Spandaw, B. van Geemen, and D. van Straten, The modularity of the Barth-Nieto quintic and its relatives, Adv. Geom. 1 (2001), no. 3, 263-289. https://doi.org/10.1515/advg.2001.017
  4. D. S. Kim, Gauss sums for $O^{-}$(2n; q), Acta Arith. 80 (1997), no. 4, 343-365.
  5. D. S. Kim,Exponential sums for $O^{-}$(2n; q) and their applications, Acta Arith. 97 (2001), no. 1, 67-86. https://doi.org/10.4064/aa97-1-4
  6. D. S. Kim, Gauss sums for symplectic groups over a finite field, Monatsh. Math. 126 (1998), no. 1, 55-71. https://doi.org/10.1007/BF01312455
  7. D. S. Kim, Exponential sums for symplectic groups and their applications, Acta Arith. 88 (1999), no. 2, 155-171. https://doi.org/10.4064/aa-88-2-155-171
  8. D. S. Kim, Infinite families of recursive formulas generating power moments of ternary Kloosterman sums with square arguments arising from symplectic groups, Adv. Math. Commun. 3 (2009), no. 2, 167-178. https://doi.org/10.3934/amc.2009.3.167
  9. D. S. Kim, Ternary codes associated with $O^{-}$(2n; q) and power moments of Kloosterman sums with square arguments, submitted.
  10. D. S. Kim, Recursive formulas generating power moments of multi-dimensional Kloosterman sums and m-multiple power moments of Kloosterman sums, submitted.
  11. D. S. Kim and J. H. Kim, Ternary codes associated with symplectic groups and power moments of Kloosterman sums with square arguments, submitted.
  12. H. D. Kloosterman, On the representation of numbers in the form $ax^{2}+by^{2}+cz^{2}+dt^{2}$, Acta Math. 49 (1927), no. 3-4, 407-464. https://doi.org/10.1007/BF02564120
  13. R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia Math. Appl. 20, Cambridge University Pless, Cambridge, 1987.
  14. R. Livne, Motivic orthogonal two-dimensional representations of Gal(Q/Q), Israel J. Math. 92 (1995), no. 1-3, 149-156. https://doi.org/10.1007/BF02762074
  15. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1998.
  16. M. Moisio, On the moments of Kloosterman sums and fibre products of Kloosterman curves, Finite Fields Appl. 14 (2008), no. 2, 515-531. https://doi.org/10.1016/j.ffa.2007.06.001
  17. C. Peters, J. Top, and M. van der Vlugt, The Hasse zeta function of a K3 surface related to the number of words of weight 5 in the Melas codes, J. Reine Angew. Math. 432 (1992), 151-176.
  18. H. Salie, Uber die Kloostermanschen Summen S(u; v; q), Math. Z. 34 (1932), no. 1, 91-109. https://doi.org/10.1007/BF01180579
  19. I. E. Shpalinski, Exponential Sums in Coding Theory and Cryptography, Lecture Notes of Tutorial Lectures given at the Institute of Mathematics of the NUS, Singapore, July 23-26, 2001.

Acknowledgement

Supported by : National Foundation of Kor