Construction of recursive formulas generating power moments of Kloosterman sums: $O^{+}(2n,2^{r})$ Case
J. Korean Math. Soc.
Published online August 22, 2019
DAE SAN KIM
Sogang University
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
$O^{+}(2n,2^{r})$. 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 $O^{+}(2n,2^{r})$.
Keywords : Kloosterman sum, 2-dimensional Kloosterman sum, orthogonal group, double cosets, maximal parabolic subgroup, Pless' power moment identity, weight distribution
MSC numbers : 11T23, 20G40, 94B05
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