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Let $p$ be an odd prime, $n$ a positive integer and $g$ a primitive root of$p^n$. Suppose$D_i^{(p^n)}=\{g^{2s+i}|s=0,1,2,\cdots,\frac{(p-1)p^{n-1}}{2}\}$, $i=0,1$, isthe generalized cyclotomic classes with $Z_{p^n}^{\ast}=D_0\cup D_1$. In thispaper, we prove that Gauss periods based on $D_0$ and $D_1$ are both equal to 0for $n\geq2$. As an application, we determine a lower bound on the 2-adiccomplexity of a class of Ding-Helleseth generalized cyclotomic sequences ofperiod $p^n$. 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 theperiod of the sequence.

A lower bound on the 2-adic complexity of Ding-Helleseth generalized cyclotomic sequences of period $p^n$