Geometric secret sharing schemes and their duals

Given a set of participants we wish to distribute information relating to a secret in such a way that only specified groups of participants can reconstruct the secret. We consider here a special class of such schemes that can be described in terms of finite geometries as first proposed by Simmons. We formalize the Simmons model and show that given a geometric scheme for a particular access structure it is possible to find another geometric scheme whose access structure is the dual of the original scheme, and which has the same average and worst-case information rates as the original scheme. In particular this shows that if an ideal geometric scheme exists then an ideal geometric scheme exists for the dual access structure.