Bounding invariants of fat points using a coding theory construction

Let Z⊆Pn be a fat point scheme, and let d(Z) be the minimum distance of the linear code constructed from Z. We show that d(Z) imposes constraints (i.e., upper bounds) on some specific shifts in the graded minimal free resolution of IZ, the defining ideal of Z. We investigate this relation in the case that the support of Z is a complete intersection; when Z is reduced and a complete intersection we give lower bounds for d(Z) that improve upon known bounds.