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

TL;DR

This work addresses the problem of characterizing entropy functions on two-dimensional faces of the polymatroidal region $Γ_4$. It extends the program from Part I by solving the remaining 10 face-types using two combinatorial design tools: mixed-level variable-strength orthogonal arrays ($MVOA$) and orthogonal Latin hypercubes, yielding new combinatorial structures. Eight faces are fully characterized and two partially characterized, with inner/outer bounds constructed via VOA/$MVOA$-based arrays. The results illuminate deep connections between entropy regions and combinatorial designs, providing a foundation for extending these techniques to higher dimensions.

Abstract

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 themself.