Codes Associated with Special Linear Groups and Power Moments of Multi-dimensional Kloosterman Sums

In this paper, we construct the binary linear codes C(SL(n, q)) associated with finite special linear groups SL(n, q), with both n,q powers of two. Then, via Pless power moment identity and utilizing our previous result on the explicit expression of the Gauss sum for SL(n, q), we obtain a recursive formula for the power moments of multi-dimensional Kloosterman sums in terms of the frequencies of weights in C(SL(n, q)). In particular, when n = 2, this gives a recursive formula for the power moments of Kloosterman sums. We illustrate our results with some examples.