Characterization of entropy functions is of fundamental importance in information theory. By imposing constraints on their Shannon outer bound, i.e., the polymatroidal region, one obtains the faces of the region and entropy functions on them with special structures. In this series of two papers, we characterize entropy functions on the $2$-dimensional faces of the polymatroidal region $Γ_4$. In Part I, we formulated the problem, enumerated all $59$ types of $2$-dimensional faces of $Γ_4$ by a algorithm, and fully characterized entropy functions on $49$ types of them. In this paper, i.e., Part II, we will characterize entropy functions on the remaining $10$ types of faces, among which $8$ types are fully characterized and $2$ types are partially characterized. To characterize these types of faces, we introduce some new combinatorial design structures which are interesting in themselves.

翻译：熵函数的表征在信息论中具有基础重要性。通过对其香农外边界（即多拟阵区域）施加约束，可获得该区域的各维面及其上具有特殊结构的熵函数。在本系列的两篇论文中，我们刻画了四维多拟阵区域 $Γ_4$ 的二维面上的熵函数。在第一部分中，我们构建了问题框架，通过算法枚举出 $Γ_4$ 所有 $59$ 类二维面，并完整刻画了其中 $49$ 类面上的熵函数。在本文（即第二部分）中，我们将刻画剩余 $10$ 类面上的熵函数，其中 $8$ 类得到完整刻画，$2$ 类获得部分刻画。为刻画这些类型的面，我们引入了一些本身具有研究价值的新组合设计结构。