  In this paper, we prove that
  (1) For any integers $n\geq 1$ and $g\geq 2$, 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 $g\geq 2$ and $n\geq 4$, there are infinitely many
non-homeomorphic closed 3-manifolds admitting distance $n$ Heegaard splittings
of genus $g$.
