An Infinite Family of Recursive Formulas Generating Power Moments of Kloosterman Sums with Trace One Arguments: O(2n+1,2^r) Case

TL;DR

This paper develops an infinite family of recursive formulas to compute odd power moments of Kloosterman sums with trace one arguments, using binary linear codes linked to orthogonal and symplectic groups over finite fields.

Contribution

It introduces a novel construction of binary linear codes associated with double cosets in orthogonal and symplectic groups, deriving recursive formulas for Kloosterman sum moments.

Findings

Recursive formulas for odd power moments of Kloosterman sums.

Explicit expressions for exponential sums over double cosets.

Connection between code weight distributions and Kloosterman sum moments.

Abstract

In this paper, we construct an infinite family of binary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the orthogonal group O(2n+1,q). Here q is a power of two. Then we obtain an infinite family of recursive formulas generating the odd power moments of Kloosterman sums with trace one arguments in terms of the frequencies of weights in the codes associated with those double cosets in O(2n+1,q) and in the codes associated with similar double cosets in the symplectic group Sp(2n,q). 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 group O(2n+1,q).