Journal of the
Korean Mathematical Society JKMS

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

MSC numbers: 11T23, 20G40, 94B05