Simple recursive formulas generating power moments of Kloosterman sums (Q2918428)

Let \(\lambda\) be the canonical additive character on a finite field \(\mathbb F_q\). The Kloosterman sum, for \(a\in\mathbb F_q^*\), is: NEWLINE\[NEWLINE K(a)=\sum_{\alpha\in\mathbb F_q^*} \lambda (\alpha +a\alpha^{-1}). NEWLINE\]NEWLINE For a positive integer \(h\), the \(h\)-th moment is: NEWLINE\[NEWLINE MK^h=\sum_{a\in\mathbb F_q^*} K(a)^h. NEWLINE\]NEWLINE Now assume that \(q=2^r\). Carlitz found \(MK^h\) for \(h\leq 4\) and \textit{M. J. Moisio} [IEEE Trans. Inf. Theory 53, No. 2, 843--847 (2007; Zbl 1177.11071)] computed it for \(h\leq 10\). Here a recursive formula for \(MK^h\) is given. The formula involves the Stirling numbers of the second kind and the weights of four binary linear codes constructed by the author.