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 of degree 4. In Part I, we formulate the problem, enumerate all 59 types of 2-dimensional faces of the region by an algorithm, and fully characterize entropy functions on 49 types of them. The entropy functions on the remaining 10 types of faces will be characterized in Part II, among which 8 types are fully characterized, and 2 types are partially characterized.

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.

In this paper, we classify all G-symmetric almost entropic regions according to their Shannon-tightness, that is, whether they can be fully characterized by Shannon-type inequalities, where$G$is a permutation group of degree 6 or 7.

In this paper, we characterize entropy functions on the 2-dimensional faces of the polymatroidal region Γ4. We enumerate all 59 types of 2-dimensional faces of Γ4and fully characterized entropy functions on 27 types of them, among which 4 types are non-trivial.