Infinite families of recursive formulas generating power moments of Kloosterman sums: O^- (2n, 2^r) case

In this paper, we construct eight infinite families of binary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group $SO^-(2n,2^r)$. Then we obtain four infinite families of recursive formulas for the power moments of Kloosterman sums and four those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of “Gauss sums” for the orthogonal groups $O^-(2n,2^r)$

💡 Research Summary

This paper investigates the deep interplay between finite orthogonal groups, binary linear codes, and the power moments of Kloosterman sums. Starting from the special orthogonal group of minus type $SO^{-}(2n,2^{r})$ over the binary field $\mathbb{F}_{2^{r}}$, the authors consider its maximal parabolic subgroup $P^{-}$ and study the double coset space $P^{-}\backslash SO^{-}(2n,2^{r})/P^{-}$. They show that this double coset space splits into eight distinct families, each of which can be identified with a set of binary vectors of length $N_i$ for $i=1,\dots,8$.

For each family $D_i$ a binary linear code $C(D_i)$ is defined by taking the characteristic function of $D_i$ as a parity‑check vector. The resulting codes have length $N_i$, dimension $k_i=N_i-\dim_{\mathbb{F}_2}P^{-}$, and their weight distributions are expressed through exponential sums of the form
\