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

In this paper, we construct three ternary linear codes associated with the orthogonal group O−(2, q) and the special orthogonal groups SO−(2, q) and SO−(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 orthogonal and special orthogonal groups O−(2n, q) and SO−(2n, q).