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In this paper, we construct two ternary linear codes associated with thesymplectic groups Sp(2,q) and Sp(4,q). Here q is a power of three. Then weobtain recursive formulas for the power moments of Kloosterman sums with squarearguments and for the 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 "Gauss sums" for the symplectic groupsSp(2n,q).

Ternary codes associated with symplectic groups and power moments of Kloosterman sums with square arguments