  In this paper, we construct two ternary linear codes associated with the
symplectic groups Sp(2,q) and Sp(4,q). Here q is a power of three. Then we
obtain 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 identity and by
utilizing the explicit expressions of "Gauss sums" for the symplectic groups
Sp(2n,q).
