Infinite Families of Recursive Formulas Generating Power Moments of Ternary Kloosterman Sums with Square Arguments Associated with $O^{-}_{}(2n,q)$

TL;DR

This paper develops recursive formulas for power moments of ternary Kloosterman sums with square arguments using code theory and exponential sums related to orthogonal groups over finite fields.

Contribution

It introduces eight infinite families of codes and derives new recursive formulas for Kloosterman sum moments using group-theoretic and coding-theoretic methods.

Findings

Four recursive formulas for power moments of Kloosterman sums with square arguments.

Four recursive formulas for even power moments of Kloosterman sums.

Explicit expressions of exponential sums over double cosets.

Abstract

In this paper, we construct eight infinite families of ternary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group $SO^{-}(2n,q)$. Here ${q}$ is a power of three. Then we obtain four infinite families of recursive formulas for power moments of Kloosterman sums with square arguments and four infinite families of recursive formulas for even power moments of those 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 exponential sums over those double cosets related to the evaluations of $"$Gauss sums" for the orthogonal groups $O^{-}(2n,q)$.