AI 中文总结
该研究证明4-熵拟阵等价于F₄可表示拟阵，结合相关结构性质完成证明，并将结论应用于理想完美访问结构的线性方案构造。
AI 中文摘要
若一个拟阵的秩函数乘以log4后，是四元字母表上随机变量的联合熵函数，则称其为4-熵拟阵。我们证明：一个拟阵是4-熵拟阵当且仅当它在F₄上可表示。该证明结合了子式闭包、四元拟阵的排除子式特征，以及四阶拟群的结构性质。因此，任意四元符号划分表示产生的拟阵均未超出四元拟阵的范畴。作为应用，每个拥有均匀四元秘密与四元活动份额的理想完美方案的访问结构，也拥有理想F₄线性方案。
英文摘要
For an integer $q\ge2$, a matroid is $q$-entropic if its rank function, multiplied by $\log q$, is the joint-entropy function of random variables on a $q$-element alphabet. We prove that a matroid is $4$-entropic if and only if it is representable over $\F_4$. The corresponding statements for alphabet sizes two and three were known. The proof combines minor closure and the excluded-minor characterization of quaternary matroids with structural properties of quasigroups of order four. Thus arbitrary four-symbol partition representations yield no matroids beyond the quaternary ones. As an application, every access structure admitting an ideal perfect scheme with a uniform four-symbol secret and four-symbol active shares also admits an ideal $\F_4$-linear scheme.
