A Representation of a Family of Secret Sharing Matroids

摘要

Deciding whether a matroid is secret sharing or not is a well-known open problem. In Ng and Walker it was shown that a matroid decomposes into uniform matroids under strong connectivity. The question then becomes as follows: when is a matroid M with N uniform components secret sharing? When N -1, M corresponds to a uniform matroid and hence is secret sharing. In this paper we show, by constructing a representation using projective geometry, that all connected matroids with two uniform components are secret sharing