L. Csirmaz:

        Secret sharing schemes on graphs

Given a graph $G$, a perfect secret sharing scheme based on $G$ is a method
to distribute a secret data among the vertices of $G$, the participants, so 
that a subset of participants can recover the secret if they contain an edge 
of $G$, otherwise they can obtain no information regarding the key. The 
average information rate is the ratio of the size of the secret and the 
average size of the share a participant must remember. The information rate 
of $G$ is the supremum of the information rates realizable by perfect secret 
sharing schemes.

Based on the entropy-theoretical arguments due to Capocelli et al, and 
extending the results of M. van Dijk, we construct a graph $G_n$ on $n$ 
vertices with average information rate below $< 4/\log n$. We obtain this 
result by determining, up to a constant factor, the average information rate 
of the $d$-dimensional cube.