Exponential Sums for O+(2n,q) and Their Applications

For a nontrivial additive character of the finite field with q elements and each positive integer r, the exponential sums ( ( trw )^r ) over w SO ⁺(2n,q) and over w O ⁺(2n,q) are considered. We show that both of them can be expressed as polynomials in q involving certain new exponential sums. Estimates on those new exponential sums are given. Also, we derive from these expressions the formulas for the number of elements w in SO ⁺(2n,q) and O ⁺(2n,q) with (trw) ^r = , for each in the finite field with q elements.