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CONSTRUCTION OF RECURSIVE FORMULAS GENERATING POWER MOMENTS OF KLOOSTERMAN SUMS: O+(2n, 2r) CASE

  • Kim, Dae San (Department of Mathematics Sogang University)
  • Received : 2019.03.18
  • Accepted : 2019.06.26
  • Published : 2020.05.01

Abstract

In this paper, we construct four infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the orthogonal group O+(2n, 2r). And we obtain two infinite families of recursive formulas for the power moments of Kloosterman sums and those of 2-dimensional Kloosterman sums 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, 2r).

Keywords

References

  1. L. Carlitz, Gauss sums over finite fields of order $2^n$, Acta Arith. 15 (1968/1969), 247-265. https://doi.org/10.4064/aa-15-3-247-265
  2. L. Carlitz, A note on exponential sums, Pacific J. Math. 30 (1969), 35-37. http://projecteuclid.org/euclid.pjm/1102978697
  3. P. Charpin, T. Helleseth, and V. Zinoviev, Propagation characteristics of $x\;\rightarrow\;x^{-1}$ and Kloosterman sums, Finite Fields Appl. 13 (2007), no. 2, 366-381. https://doi.org/10.1016/j.ffa.2005.08.007
  4. J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coeffcients of cusp forms, Invent. Math. 70 (1982/83), no. 2, 219-288. https://doi.org/10.1007/BF01390728
  5. H. Dobbertin, P. Felke, T. Helleseth, and P. Rosendahl, Niho type cross-correlation functions via Dickson polynomials and Kloosterman sums, IEEE Trans. Inform. Theory 52 (2006), no. 2, 613-627. https://doi.org/10.1109/TIT.2005.862094
  6. R. Evans, Seventh power moments of Kloosterman sums, Israel J. Math. 175 (2010), 349-362. https://doi.org/10.1007/s11856-010-0014-0
  7. 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
  8. 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
  9. 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
  10. D. S. Kim, Codes associated with special linear groups and power moments of multidimensional Kloosterman sums, Ann. Mat. Pura Appl. (4) 190 (2011), no. 1, 61-76. https://doi.org/10.1007/s10231-010-0138-1
  11. D. S. Kim, Codes associated with $O^+(2n,\;2^r)$ and power moments of Kloosterman sums, Integers 12 (2012), no. 2, 237-257. https://doi.org/10.1515/integ.2011.100
  12. D. S. Kim and Y. H. Park, Gauss sums for orthogonal groups over a finite field of characteristic two, Acta Arith. 82 (1997), no. 4, 331-357. https://doi.org/10.4064/aa-82-4-331-357
  13. 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
  14. G. Lachaud and J. Wolfmann, The weights of the orthogonals of the extended quadratic binary Goppa codes, IEEE Trans. Inform. Theory 36 (1990), no. 3, 686-692. https://doi.org/10.1109/18.54892
  15. R. Lidl and H. Niederreiter, Finite Fields, second edition, Encyclopedia of Mathematics and its Applications, 20, Cambridge University Press, Cambridge, 1997.
  16. R. Livne, Motivic orthogonal two-dimensional representations of Gal($\bar-Q}$/Q), Israel J. Math. 92 (1995), no. 1-3, 149-156. https://doi.org/10.1007/BF02762074
  17. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Publishing Co., Amsterdam, 1977.
  18. M. J. Moisio, The moments of a Kloosterman sum and the weight distribution of a Zetterberg-type binary cyclic code, IEEE Trans. Inform. Theory 53 (2007), no. 2, 843-847. https://doi.org/10.1109/TIT.2006.889020
  19. 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.
  20. H. Salie, Uber die Kloostermanschen Summen S(u, v; q), Math. Z. 34 (1931), 91-109.
  21. R. Schoof and M. van der Vlugt, Hecke operators and the weight distributions of certain codes, J. Combin. Theory Ser. A 57 (1991), no. 2, 163-186. https://doi.org/10.1016/0097-3165(91)90016-A