  In this paper, we construct two binary linear codes associated with
multi-dimensional and $m -$multiple power Kloosterman sums (for any fixed $m$)
over the finite field $\mathbb{F}_{q}$. Here $q$ is a power of two. The former
codes are dual to a subcode of the binary hyper-Kloosterman code. Then we
obtain two recursive formulas for the power moments of multi-dimensional
Kloosterman sums and for the $m$-multiple power moments of Kloosterman sums in
terms of the frequencies of weights in the respective codes. This is done via
Pless power moment identity and yields, in the case of power moments of
multi-dimensional Kloosterman sums, much simpler recursive formulas than those
associated with finite special linear groups obtained previously.
