  Let $p$ be an odd prime, $n$ a positive integer and $g$ a primitive root of
$p^n$. Suppose
$D_i^{(p^n)}=\{g^{2s+i}|s=0,1,2,\cdots,\frac{(p-1)p^{n-1}}{2}\}$, $i=0,1$, is
the generalized cyclotomic classes with $Z_{p^n}^{\ast}=D_0\cup D_1$. In this
paper, we prove that Gauss periods based on $D_0$ and $D_1$ are both equal to 0
for $n\geq2$. As an application, we determine a lower bound on the 2-adic
complexity of a class of Ding-Helleseth generalized cyclotomic sequences of
period $p^n$. The result shows that the 2-adic complexity is at least
$p^n-p^{n-1}-1$, which is larger than $\frac{N+1}{2}$, where $N=p^n$ is the
period of the sequence.
