Codes Associated with $O(3,2^r)$ and Power Moments of Kloosterman Sums with Trace One Arguments

TL;DR

This paper constructs a binary linear code related to the orthogonal group over a binary field and derives a recursive formula for odd power moments of Kloosterman sums with trace one arguments, linking coding theory and exponential sum analysis.

Contribution

It introduces a new code associated with $O(3,q)$ and establishes a recursive relation for Kloosterman sum moments using Pless power moment identity and Gauss sums.

Findings

Derived recursive formulas for odd power moments of Kloosterman sums.

Connected code weight distributions with exponential sum moments.

Utilized explicit Gauss sum expressions for orthogonal groups.

Abstract

We construct a binary linear code $C(O(3,q))$, associated with the orthogonal group $O(3,q)$. Here $q$ is a power of two. Then we obtain a recursive formula for the odd power moments of Kloosterman sums with trace one arguments in terms of the frequencies of weights in the codes $C(O(3,q))$ and $C(Sp(2,q))$. This is done via Pless power moment identity and by utilizing the explicit expressions of Gauss sums for the orthogonal groups.