  Given an Artinian local ring $R$, we define its Gorenstein colength $g(R)$ to
measure how closely we can approximate $R$ by a Gorenstein Artin local ring. In
this paper, we show that $R = T/I$ satisfies the inequality $g(R) \leq
\lambda(R/\soc(R))$ in the following two cases: (a) $T$ is a power series ring
over a field of characteristic zero and $I$ an ideal that is the power of a
system of parameters or (b) $T$ is a 2-dimensional regular local ring with
infinite residue field and $I$ is primary to the maximal ideal of $T$.
  In the first case, we compute $g(R)$ by constructing a Gorenstein Artin local
ring mapping onto $R$. We further use this construction to show that an ideal
that is the $n$th power of a system of parameters is directly linked to the
$(n-1)$st power via Gorenstein ideals. A similar method shows that such ideals
are also directly linked to themselves via Gorenstein ideals.
  Keywords: Gorenstein colength; Gorenstein linkage.
