Ternary codes associated with \(\mathrm{O}(3,3^r)\) and power moments of Kloosterman sums with trace nonzero square arguments (Q2915737)

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 of the Kloosterman sum with trace nonzero square arguments is: NEWLINE\[NEWLINE T_{12}SK^h =\sum_{a\in \mathbb F_q^*, tr a\neq 0} K(a^2)^h. NEWLINE\]NEWLINE Now assume that \(q=3^r\). The author gives recursive formulas for \(T_{12}SK^h\). The formulas involve the Stirling numbers of the second kind and the weights of two ternary linear codes, one constructed from \(\mathrm{SO}(3,q)\) and the other from \(\mathrm{O}(3,q)\).