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 [4], and extending the results of M. van Dijk [7] and Blundo et al [2], we construct a graph Gn 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.