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In this paper, we construct eight infinite families of ternary linear codesassociated with double cosets with respect to certain maximal parabolicsubgroup of the special orthogonal group $SO^{-}(2n,q)$. Here ${q}$ is a powerof three. Then we obtain four infinite families of recursive formulas for powermoments of Kloosterman sums with square arguments and four infinite families ofrecursive formulas for even power moments of those in terms of the frequenciesof weights in the codes. This is done via Pless power moment identity and byutilizing the explicit expressions of exponential sums over those double cosetsrelated to the evaluations of $"$Gauss sums" for the orthogonal groups$O^{-}(2n,q)$.

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