The Heegaard distances cover all non-negative integers

In this paper, we prove that (1) For any integers $n\ge 1$ and $g\ge 2$, there is a closed 3-manifold $M_g^n$ which admits a distance $n$ Heegaard splitting of genus $g$ except that the pair of $(g,n)$ is $(2,1)$. Furthermore, $M_g^n$ can be chosen to be hyperbolic except that the pair of $(g,n)$ is $(3,1)$. (2) For any integers $g\ge 2$ and $n\ge 4$, there are infinitely many non-homeomorphic closed 3-manifolds admitting distance $n$ Heegaard splittings of genus $g$.