Infinite Families of Recursive Formulas Generating Power Moments of Kloosterman Sums: O\^{+}(2n, 2\^{r}) Case

TL;DR

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

Contribution

It introduces four infinite families of codes linked to orthogonal groups and derives recursive formulas for Kloosterman sum moments using these codes.

Findings

Derived recursive formulas for Kloosterman sum moments.

Constructed new families of binary linear codes.

Connected code weight distributions to exponential sum evaluations.

Abstract

In this paper, we construct four infinite families of binary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the orthogonal group O^+(2n,2^r). Here q is a power of two. Then we obtain two infinite families of recursive formulas for the power moments of Kloosterman sums and 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).