Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-19T05:02:01.374Z Has data issue: false hasContentIssue false
  • English
  • Français

CODES ASSOCIATED WITH Sp(4,q) AND EVEN-POWER MOMENTS OF KLOOSTERMAN SUMS

Part of: Exponential sums and character sums Communication, information

Published online by Cambridge University Press:  17 April 2009

JI HYUN KIM
JI HYUN KIM*
Affiliation:
Department of Mathematics, Sogang University, Seoul 121–742, South Korea (email: mip97@hanmail.net)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Here we derive a recursive formula for even-power moments of Kloosterman sums or equivalently for power moments of two-dimensional Kloosterman sums. This is done by using the Pless power-moment identity and an explicit expression of the Gauss sum for Sp(4,q).

Keywords

linear codesymplectic groupGauss sumKloosterman sumPless power moment identityweight distribution

MSC classification

Secondary: 94A24: Coding theorems (Shannon theory) 11L05: Gauss and Kloosterman sums; generalizations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 3 , June 2009 , pp. 427 - 435
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This work was supported by grant No. R01-2006-000-11176-0 from the Basic Research Program of the Korea Science and Engineering Foundation.

References

[1] Carlitz, L., ‘Gauss sums over finite fields of order 2n’, Acta Arith. 15 (1969), 247–265.CrossRefGoogle Scholar
[2] Carlitz, L., ‘A note on exponential sums’, Pacific J. Math. 30 (1969), 35–37.Google Scholar
[3] Kim, D. S., ‘Gauss sums for symplectic groups over a finite field’, J. Monatsh. Math. 126 (1998), 55–71.CrossRefGoogle Scholar
[4] Kim, D. S., ‘Codes associated with special linear groups and power moments of multi-dimensional Kloosterman sums’, submitted.Google Scholar
[5] Kloosterman, H. D., ‘On the representation of numbers in the form ax 2+by 2+cz 2+dt 2’, Acta Math. 49 (1926), 407–464.CrossRefGoogle Scholar
[6] Lachaud, G. and Wolfmann, J., ‘The weights of the orthogonals of the extended quadratic binary Goppa codes’, IEEE Trans. Inform. Theory 36 (1990), 686–692.Google Scholar
[7] Lidl, R. and Niederreiter, H., Finite Fields, 2nd edn, Encyclopedia of Mathematics and its Applications, 20 (Cambridge University Press, Cambridge, UK, 1997).Google Scholar
[8] MacWilliams, F. J. and Sloane, N. J. A., The Theory of Error Correcting Codes (North-Holland, Amsterdam, The Netherlands, 1998).Google Scholar
[9] Moisio, M. J., ‘The moments of a Kloosterman sum and the weight distribution of a Zetterberg-type binary cyclic code’, IEEE Trans. Inform. Theory 53(2) (2007), 843–847.Google Scholar
[10] Moisio, M., ‘Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm’, Acta Arith. 132(4) (2008), 329–350.Google Scholar
[11] Moisio, M. and Ranto, K., ‘Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros’, Finite Fields Appl. 13 (2007), 922–935.CrossRefGoogle Scholar
[12] Schoof, R. and van der Vlugt, M., ‘Hecke operators and the weight distributions of certain codes’, J. Combin. Theory Ser. A 57 (1991), 163–186.Google Scholar