We generalize the Diederich-Fornaess index to bounded weakly q-convex domains with bounded q-subharmonic exhaustion functions. Sufficient conditions for this generalized Diederich-Fornæss index to have a given lower bound are proved. We show this generalized index is positive on bounded weakly q-convex domains with C^3 boundaries. Additionally, we prove sufficient conditions for this generalized index to equal one. For example, we show that if the domain has Property ( ̃(Pq ) ) then the domain has high hyperconvexity
