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

翻译：$k$-一致超图是指每个 $k$-超边恰好包含 $k$ 个顶点的超图。$k$-均匀访问结构由 $k$-一致超图 $\mathcal{H}$ 表示，其中参与者对应超图 $\mathcal{H}$ 的顶点。若一组顶点通过 $k$-超边相连，则可从其秘密份额重构秘密值；而一组不相邻的顶点则无法获取任何秘密信息。衡量秘密共享方案效率的参数之一是信息率，定义为秘密长度与分配给参与者的最大份额长度之比。信息率等于1的秘密共享方案称为理想秘密共享方案。若存在理想秘密共享方案可实现某访问结构，则该访问结构被视为理想访问结构。刻画理想访问结构是秘密共享方案中的重要问题之一，许多学者已对这一课题展开研究~\cite{BD, CT,JZB, FP1,FP2,DS1,TD}。本文利用独立序列方法刻画了理想 $k$-均匀访问结构。特别地，我们证明：当 $\Gamma$ 的最优信息率大于 $\frac{k-1}{k}$ 时，$\Gamma$ 的约化访问结构为 $(k, n)$-门限访问结构，其中 $\Gamma$ 为满足特定条件的 $k$-均匀访问结构。