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In this paper, we construct two binary linear codes associated withmulti-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 formercodes are dual to a subcode of the binary hyper-Kloosterman code. Then weobtain two recursive formulas for the power moments of multi-dimensionalKloosterman sums and for the $m$-multiple power moments of Kloosterman sums interms of the frequencies of weights in the respective codes. This is done viaPless power moment identity and yields, in the case of power moments ofmulti-dimensional Kloosterman sums, much simpler recursive formulas than thoseassociated with finite special linear groups obtained previously.

Recursive formulas generating power moments of multi-dimensional Kloosterman sums and $m$-multiple power moments of Kloosterman sums