Four-Entropic Matroids Are Quaternary

A matroid is $4$-entropic if its rank function, multiplied by $\log 4$, is the joint-entropy function of random variables on a four-element alphabet. We prove that a matroid is $4$-entropic if and only if it is representable over ${\mathbb{F}}_4$. 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 ${\mathbb{F}}_4$-linear scheme.