Recursive formulas generating power moments of multi-dimensional Kloosterman sums and $m$-multiple power moments of Kloosterman sums

TL;DR

This paper develops recursive formulas for calculating power moments of multi-dimensional and m-multiple Kloosterman sums over finite fields of characteristic two, simplifying previous methods using coding theory techniques.

Contribution

It introduces new recursive formulas for Kloosterman sum moments derived from binary linear codes, improving upon earlier approaches with more straightforward recursive relations.

Findings

Derived recursive formulas for multi-dimensional Kloosterman sum moments

Established connections between Kloosterman sums and binary linear codes

Provided simpler recursive formulas compared to previous methods

Abstract

In this paper, we construct two binary linear codes associated with multi-dimensional and $m -$multiple power Kloosterman sums (for any fixed $m$) over the finite field $\mathbb{F}_{q}$. Here $q$ is a power of two. The former codes are dual to a subcode of the binary hyper-Kloosterman code. Then we obtain two recursive formulas for the power moments of multi-dimensional Kloosterman sums and for the $m$-multiple power moments of Kloosterman sums in terms of the frequencies of weights in the respective codes. This is done via Pless power moment identity and yields, in the case of power moments of multi-dimensional Kloosterman sums, much simpler recursive formulas than those associated with finite special linear groups obtained previously.