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

The 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 Γ4. In Part I, we formulated the problem, enumerated all 59 types of 2-dimensional faces of Γ4by an 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 that are interesting in themselves.