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

Abstract: Let $p$ be an odd prime, $n$ a positive integer and $g$ a primitive root of $p<sup>n$. Suppose $$D_i<sup>{(p<sup>n)}={g<sup>{2s+i}|s=0,1,2,\cdots,\frac{(p-1)p<sup>{n-1}}{2}}$$, $i=0,1$, 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 $p<sup>n$. The result shows that the 2-adic complexity is at least $p<sup>n-p<sup>{n-1}-1$, which is larger than $\frac{N+1}{2}$, where $N=p<sup>n$ is the period of the sequence.