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In this paper, we construct two infinite families of ternary linear codes associated with double cosets with respect to certain maximal parabolic subgroups of the symplectic group $S_p(2n, q)$. Here $q$ is a power of three. Then we obtain infinite families of recursive formulas for the power moments of Kloosterman sums with square arguments and for the even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identities and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of ''Gauss sums'' for the symplectic groups $S_p(2n, q)$.

Infinite families of recursive formulas generating power moments of ternary Kloosterman sums with square arguments arising from symplectic groups