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

TL;DR

This paper develops new recursive formulas for power moments of Kloosterman sums using binary linear codes associated with double cosets in special orthogonal groups, expanding understanding of exponential sums in finite fields.

Contribution

It introduces eight infinite families of codes and derives four recursive formulas each for Kloosterman and 2-dimensional Kloosterman sums, linking coding theory with exponential sum analysis.

Findings

Four recursive formulas for Kloosterman sum moments.

Four recursive formulas for 2-dimensional Kloosterman sum moments.

Explicit expressions of exponential sums over double cosets.

Abstract

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 $O^-(2n,2^r)$