Let I be a Cohen-Macaulay ideal of grade g > 0 in a local Gorenstein ring (R, m) with residue class field k. An R-ideal J is said to be linked to I with respect to the regular sequence α = α1 …, αg ⊂ I ∩ J if J = (α): I and I = (α): J ([6]). In this paper we are concerned with the following question: how big is dimk ((α, mJ)/mJ)? Obviously this dimension is at most g, but it could be as small as 0. If it is g then the link from J to I is called a minimal link, which is in most respects the desired type of link. The only general result known in this direction is that if I is Gorenstein, then dimk {(α, mJ)/mJ) = g unless both I and J are complete intersections (see [1], Proposition 5.2). We are able to generalize this fact to the case where (R/I)p is Gorenstein for all prime ideals p in R/I with dim (R/J)p ≤ 4; however we have to assume that I is generically a complete intersection ideal, and that R is a complete intersection (Theorem 2.3). Without the assumption on R we prove that if I is generically a complete intersection, and if for a fixed integer r the type of (R/I)p is at most r for all prime ideals p in R/I with dim (R/I)P ≤ (r + 1)2, then dimk ((α, mJ/mJ)) ≥ g — r (Proposition 2.1). If r = 1, i.e. if R/I is Gorenstein in codimension 4, then this estimate shows the dimension is at least g — 1. Theorem 2.3 can also be interpreted to yield a strong upper bound for the codimension of the non-Gorenstein-locus of certain perfect ideals: Let R be a regular local ring. Let I be an R-ideal which is generically a complete intersection, and assume that I is in the even linkage class of a Gorenstein ideal (i.e., there exists a sequence of links I ~ I1 ~ I2 ~ … ~ I2n with I2n a Gorenstein ideal); then I is a Gorenstein ideal provided that {R/I)p is Gorenstein for all prime ideals p of R/I with dim (R/I)p ≤ 4 (Corollary 3.1).