  In a secret sharing scheme, shares of a secret are distributed to
participants in such a way that only certain predetermined sets of participants
are qualified to reconstruct the secret. An access structure on a set of
participants specifies which sets are to be qualified. The information rate of
an access structure is a bound on how efficient a secret sharing scheme for
that access structure can be. Marti-Farre and Padro showed that all access
structures with information rate greater than two-thirds are matroid-related,
and Stinson showed that four of the minor-minimal, non-matroid-related access
structures have information rate exactly two-thirds. By a result of Seymour,
there are infinitely many remaining minor-minimal, non-matroid-related access
structures. In this paper we find the exact information rates for all such
structures.
