Define $g(n,m)$ to be the largest integer $k$ such that $2^k$ divides $(\genfrac{}{}{0ex}{}{n}{m})$. For example, $(\genfrac{}{}{0ex}{}{12}{5})=792=2^3\cdot 3^2\cdot 11$, hence $g(12,5)=3$. Then define $F(n)=max\{g(n,m):0\le m\le n\}$. $F(10)=3$ and $F(100)=6$.

Let $S(N)$ = ${\displaystyle \sum _{n=1}^{N}F(n)}$. You are given that $S(100)=389$ and $S({10}^7)=203222840$.

Find $S({10}^{16})$.