CN-SMO 2017 Problem 1

Let $x_i \in \{0, 1\} (i = 1, 2, \cdots, n)$ . If the function $f = f(x_1, x_2, \cdots, x_n)$ only equals $0$ or $1$ , then define $f$ as an " $n$ -variable Boolean function" and denote $$D_n (f) = \{ (x_1, x_2, \cdots, x_n) | f(x_1, x_2, \cdots, x_n) = 0 \}$$ . $(1)$ Determine the number of $n$ -variable Boolean functions; $(2)$ Let $g$ be a $10$ -variable Boolean function satisfying $$g(x_1, x_2, \cdots, x_{10}) \equiv 1 + x_1 + x_1 x_2 + x_1 x_2 x_3 + \cdots + x_1 x_2\cdots x_{10} \pmod{2}$$ Evaluate the size of the set $D_{10} (g)$ and $\sum\limits_{(x_1, x_2, \cdots, x_{10}) \in D_{10} (g)} (x_1 + x_2 + x_3 + \cdots + x_{10})$ .