Abstract: Let p be an odd prime, n,e,m and r be integers satisfying the following conditions:n=em,1 ≤ r ≤ e-1,(r,e)=1. Let k be a complex primitive pth root of unity and J:z→k^trm₁(z) be the nontrivial addition character of the finite field F_q(q=p^m). For every pair a,b∈F_q^e,Let e_(a,b) denote the exponential sum:e_(a,b)=∑_x∈Fq^eJ[tr_mⁿ(ax+bx^qr+1)], where tr_mⁿ(°) denotes the trace function of F_q^e over F_q. In this paper we determined the values of the exponential sum e_(a,b) and discussed the applications of exponential sums in spreading sequences designs.

Key words: exponential sums, spreading sequences, quadratic forms