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

Let $p$ be an odd prime, $n$ a positive integer and $g$ a primitive root of $p^n$. Suppose $Di^{{(pn)}={g{2s+i}|s=0,1,2,\cdots,\frac{(p-1)p{n-1}}{2}}$,} $i=0,1$, is the generalized cyclotomic classes with $Z{p^{n}{\ast}=D_0\cup} D1$. In this paper, we prove that Gauss periods based on $D0$ 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.