Codes associated with orthogonal groups and power moments of Kloosterman sums (Q2869650)

Let \(\lambda\) be the canonical additive character on a finite field \(\mathbb F_q\). The \(m\)-dimensional Kloosterman sum, for \(a\in \mathbb F_q^*\), is: NEWLINE\[NEWLINE K_m(a)=\sum_{\alpha_1, \ldots ,\alpha_m\in\mathbb F_q^*} \lambda (\alpha_1+\cdots +\alpha_m +a\alpha_1^{-1}\cdots\alpha_m^{-1}). NEWLINE\]NEWLINE For a positive integer \(h\), the \(h\)-th moment of the Kloosterman sum is: NEWLINE\[NEWLINE MK_m^h =\sum_{a\in \mathbb F_q^*} K_m(a)^h. NEWLINE\]NEWLINE Now assume that \(q=2^r\). The author gives recursive formulas for \(MK_1^h\) and \(MK_2^h\). The formulas involve the Stirling numbers of the second kind and the weights of three binary linear codes, constructed from \(\mathrm{SO}_2(\theta ,q)\), \(\mathrm{O}_2(\theta , q)\) and \(\mathrm{SO}_4(\theta,q)\), where \(\theta\) is the quadratic form \(\sum_{i=1}^{n-1} x_ix_{n-1+i}+x_{2n-1}^2+x_{2n-1}x_{2n}+bx_{2n}^2\), \(b\in \mathbb F_q\setminus\{\alpha^2+\alpha : \alpha\in\mathbb F_q\}\).