Entropy Functions on Two-Dimensional Faces of Polymatroidal Region of Degree Four: Part II: Information Theoretic Constraints Breed New Combinatorial Structures

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 ${\mathrm{\Gamma }}_4$. In Part I, we formulated the problem, enumerated all $59$ types of $2$-dimensional faces of ${\mathrm{\Gamma }}_4$ 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.