Ternary Codes Associated with $O(3,3^r)$ and Power Moments of Kloosterman Sums with Trace Nonzero Square Arguments

TL;DR

This paper constructs ternary linear codes linked to orthogonal groups over finite fields of characteristic three and derives recursive formulas for power moments of Kloosterman sums with specific trace conditions.

Contribution

It introduces new ternary codes associated with $SO(3,q)$ and $O(3,q)$ and establishes recursive formulas for Kloosterman sum moments using these codes.

Findings

Recursive formulas for Kloosterman sum moments derived

Explicit expressions of Gauss sums for orthogonal groups used

Codes constructed for orthogonal groups over finite fields of characteristic three

Abstract

In this paper, we construct two ternary linear codes $C(SO(3,q))$ and $C(O(3,q))$, respectively associated with the orthogonal groups $SO(3,q)$ and $O(3,q)$. Here $q$ is a power of three. Then we obtain two recursive formulas for the power moments of Kloosterman sums with $``$trace nonzero square arguments" 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 Gauss sums for the orthogonal groups.