  In this paper, we construct three binary linear codes $C(SO^{-}(2,q))$,
$C(O^{-}(2,q))$, $C(SO^{-}(4,q))$, respectively associated with the orthogonal
groups $SO^{-}(2,q)$, $O^{-}(2,q)$, $SO^{-}(4,q)$, with $q$ powers of two. Then
we obtain recursive formulas for the power moments of Kloosterman and
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 Gauss sums for the orthogonal groups. We emphasize
that, when the recursive formulas for the power moments of Kloosterman sums are
compared, the present one is computationally more effective than the previous
one constructed from the special linear group $SL(2,q)$. We illustrate our
results with some examples.
