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In this paper, we construct three ternary linear codes associated with theorthogonal group O^-(2,q) and the special orthogonal groups SO^-(2,q) andSO^-(4,q). Here q is a power of three. Then we obtain recursive formulas forthe power moments of Kloosterman sums with square arguments and for the evenpower 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 explicitexpressions of "Gauss sums" for the orthogonal and special orthogonal groupsO^-(2n,q) and SO^-(2n,q).

Ternary codes associated with O − (2n, q) and power moments of kloosterman sums with square arguments*