The bicanonical map of surfaces with $p_g=0$ and $K^2\ge 7$

A minimal surface of general type with $p_g(S)=0$ satisfies $1\le K^2\le 9$ and it is known that the image of the bicanonical map $\text{fie}$ is a surface for $K_S^2\ge 2$, whilst for $K_S^2\ge 5$, the bicanonical map is always a morphism. In this paper it is shown that $\text{fie}$ is birational if $K_S^2=9$ and that the degree of $\text{fie}$ is at most 2 if $K_S^2=7$ or $K_S^2=8$. By presenting two examples of surfaces $S$ with $K_S^2=7$ and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with $K_S^2=8$ is, to our knowledge, a new example of a surface of general type with $p_g=0$.