CN-SMO 2017 Problem 2

Let $x_i \in \{0,1\}(i=1,2,\cdots ,n)$ ,if the value of function $f=f(x_1,x_2, \cdots ,x_n)$ can only be $0$ or $1$ ,then we call $f$ a $n$ -var Boole function,and we denote $D_n(f)=\{(x_1,x_2, \cdots ,x_n)|f(x_1,x_2, \cdots ,x_n)=0\}.$ $(1)$ Find the number of $n$ -var Boole function; $(2)$ Let $g$ be a $n$ -var Boole function such that $g(x_1,x_2, \cdots ,x_n) \equiv 1+x_1+x_1x_2+x_1x_2x_3 +\cdots +x_1x_2 \cdots x_n \pmod 2$ , Find the number of elements of the set $D_n(g)$ ,and find the maximum of $n \in \mathbb{N}_+$ such that $\sum_{(x_1,x_2, \cdots ,x_n) \in D_n(g)}(x_1+x_2+ \cdots +x_n) \le 2017.$