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A $k$-uniform hypergraph is a hypergraph where each $k$-hyperedge has exactly$k$ vertices. A $k$-homogeneous access structure is represented by a$k$-uniform hypergraph $\mathcal{H}$, in which the participants correspond tothe vertices of hypergraph $\mathcal{H}$. A set of vertices can reconstruct thesecret value from their shares if they are connected by a $k$-hyperedge, whilea set of non-adjacent vertices does not obtain any information about thesecret. One parameter for measuring the efficiency of a secret sharing schemeis the information rate, defined as the ratio between the length of the secretand the maximum length of the shares given to the participants. Secret sharingschemes with an information rate equal to one are called ideal secret sharingschemes. An access structure is considered ideal if an ideal secret sharingscheme can realize it. Characterizing ideal access structures is one of theimportant problems in secret sharing schemes. The characterization of idealaccess structures has been studied by many authors~\cite{BD, CT,JZB,FP1,FP2,DS1,TD}. In this paper, we characterize ideal $k$-homogeneous accessstructures using the independent sequence method. In particular, we prove thatthe reduced access structure of $\Gamma$ is an $(k, n)$-threshold accessstructure when the optimal information rate of $\Gamma$ is larger than$\frac{k-1}{k}$, where $\Gamma$ is a $k$-homogeneous access structuresatisfying specific criteria.

On Ideal Secret-Sharing Schemes for $k$-homogeneous access structures