A Recursive Formula for Power Moments of 2-Dimensional Kloosterman Sums Assiciated with General Linear Groups

TL;DR

This paper develops a recursive formula for calculating the power moments of 2-dimensional Kloosterman sums associated with the general linear group over finite fields of characteristic two, using coding theory and Pless identities.

Contribution

It introduces a new recursive approach linking Kloosterman sum moments with weight distributions of a specially constructed binary linear code.

Findings

Derived recursive formula for power moments of 2D Kloosterman sums.

Connected Kloosterman sums with code weight distributions.

Utilized Pless power moment identity and explicit sum expressions.

Abstract

In this paper, we construct a binary linear code connected with the Kloosterman sum for $GL(2,q)$. Here $q$ is a power of two. Then we obtain a recursive formula generating the power moments 2-dimensional Kloosterman sum, equivalently that generating the even power moments of Kloosterman sum in terms of the frequencies of weights in the code. This is done via Pless power moment identity and by utilizing the explicit expression of the Kloosterman sum for $GL(2,q)$.