Secret sharing schemes on sparse homogeneous access structures with rank three. Electronic Journal of Combinatorics 11 (1) (2004) Research Paper 72, 16 pp. (electronic).

One of the main open problems in secret sharing is the characterization of the ideal access structures. This problem has been studied for several families of access structures with similar results. Namely, in all these families, the ideal access structures coincide with the vector space ones and, besides, the optimal information rate of a non-ideal access structure is at most $2/3$.

A first approach to the solution of that problem for the family of the $3$-homogeneous access structures is made in this paper. First, we present an ideal $3$-homogeneous access structure that is not vector space. Afterwards, we prove that the $3$-homogeneous access structures that can be realized by a $\mathbb{Z}_2$-vector space secret sharing scheme are sparse, that is, any subset of four participants contains at most two minimal qualified subsets. Finally, we solve the characterization problem for the family of the sparse $3$-homogeneous access structures. Specifically, we completely characterize the ideal access structures in this family, we prove that they coincide with the $\mathbb{Z}_2$-vector space ones and, besides, we demonstrate that there is no structure in this family having optimal information rate between $2/3$ and $1$. That is, we establish that the properties that were previously proved for several families also hold for the  family of the sparse $3$-homogeneous access structures.

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