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 $Γ_4$. 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.

💡 Research Summary

This paper continues a two‑part series on the characterization of entropy functions that lie on two‑dimensional faces of the polymatroidal region Γ₄ (the Shannon outer bound for four random variables). Part I enumerated all 59 distinct types of 2‑dimensional faces of Γ₄ and completely characterized entropy functions on 49 of them. The present work tackles the remaining ten face types.

Background.
For a random vector X = (X₁,…,Xₙ) the set function h(A)=H(X_A) is called an entropy function. Entropy functions satisfy the polymatroid axioms (non‑negativity, monotonicity, submodularity) and thus belong to the polyhedral cone Γₙ, the Shannon outer bound. The true entropy region Γₙ is a proper subset of Γₙ when n≥4, because of non‑Shannon information inequalities (e.g., Zhang‑Yeung). Each Shannon‑type inequality defines a facet of Γₙ; intersecting a collection of facets yields a face F. The set F = F ∩ Γₙ contains all entropy functions that satisfy the equalities defining F. Characterizing F for a given face is the central problem of this line of research.

Structure of Γ₄.
When n=4, Γ₄ has 28 facets (the elemental inequalities) and 41 extreme rays. The extreme rays fall into 11 families (uniform matroids, wheel matroids, special polymatroids denoted ˆU, and the V‑amos polymatroid). Each family can be represented by a minimal integer polymatroid whose rank vector is listed in Table 1 of the paper.

New combinatorial tools.
The ten unresolved faces involve extreme rays whose integer polymatroids have ranks larger than 1, for which the classical matroid‑based approach is insufficient. The authors therefore introduce two families of combinatorial designs:

1. Mixed‑level variable‑strength orthogonal arrays (MV‑OA).
   Given an integer polymatroid P with rank function r and a base level v>1, an MV‑OA is a v^{r(N)} × n array whose column i takes values from a set of size v^{r(i)}‑1. For any subset A⊆N, the subarray T(A) contains each possible row exactly v^{r(N)‑r(A)} times. This generalizes ordinary orthogonal arrays (OA) and variable‑strength OAs (V‑OA) that arise from uniform matroids.

2. Orthogonal Latin hypercubes.
   These are higher‑dimensional analogues of Latin squares. For the uniform matroid U_{2,3}, a V‑OA corresponds to a Latin square of order v (where rows, columns, and symbols are all distinct). The authors also define “zeroth‑kind” (all entries zero) and “second‑kind” (symbols drawn from a larger alphabet) Latin squares, which serve as building blocks for more complex designs.

The two frameworks are essentially equivalent, but MV‑OA makes the symmetry among variables explicit and scales more naturally to higher n, while Latin hypercubes provide a clear visual representation for the four‑variable case.

Characterization of the ten faces.
Each of the ten faces is spanned by two extreme rays. The authors select a representative extreme ray from each family and construct the corresponding MV‑OA or Latin hypercube. The main correspondences are:

• Face spanned by U_{123}^{2,3} and U_{4}^{1,1}.
  The MV‑OA T has columns T(1), T(2), T(3) forming a V‑OA for U_{2,3} (a standard Latin square), while column T(4) is constant. This yields a pair of Latin squares sharing the same row/column indices: one is a “first‑kind” square (all symbols distinct) and the other a “zeroth‑kind” square (all zeros).

• Face spanned by the wheel matroid W_{34}.
  The array T satisfies T(1,2,3) = VOA(U_{2,3}, v) and T(4)=T(3). Hence the two Latin squares are identical.

• Face spanned by U_{2,4}.
  The corresponding OA is a pair of mutually orthogonal Latin squares (MOLS) of order v.

• Faces involving the polymorphisms ˆU_{i}^{2,5} and ˆU_{i}^{3,5}.
  These are obtained via free expansions of the uniform matroids U_{2,5} and U_{3,5}. Their MV‑OAs combine rows of different strengths and give rise to more intricate hyper‑cubic structures.

• Face spanned by V_{α}^{8}.
  The V‑amos matroid leads to an MV‑OA where each row encodes a three‑dimensional orthogonal array; the associated Latin hypercube can be viewed as a 3‑dimensional Latin cube with additional strength constraints.

Through these constructions the authors prove that an entropy function lies on a given face F iff the associated MV‑OA (or equivalently the Latin hypercube) exists. Consequently, the feasible region F* is either a line segment (when the design is unique) or a higher‑dimensional polytope (when multiple designs are possible).

Partial characterizations.
Two of the ten faces resist a full description using the present designs. For these, the authors show that the known MV‑OAs provide necessary conditions but not sufficient ones. They conjecture that additional non‑Shannon information inequalities or yet‑unknown combinatorial objects (e.g., non‑regular mixed‑strength arrays) are required to close the gap.

Implications and future directions.
By completing the analysis of all 2‑dimensional faces of Γ₄, the paper establishes a concrete bridge between information‑theoretic constraints and combinatorial design theory. The introduced MV‑OAs and orthogonal Latin hypercubes constitute new families of designs motivated directly by entropy considerations. This synergy suggests several avenues for further research:

• Extending the MV‑OA framework to n>4 and to higher‑dimensional faces (3‑D, 4‑D, …).
• Investigating the role of the newly discovered designs in network coding, secret sharing, distributed storage, and coded caching, where the underlying constraints often define a face of Γₙ.
• Exploring whether the partial‑characterized faces can be fully resolved by discovering new non‑Shannon inequalities or by constructing novel mixed‑strength arrays.

In summary, the paper not only finishes the classification of entropy functions on the 2‑dimensional faces of Γ₄ but also uncovers a rich interplay between information theory and combinatorial design, opening a promising research frontier.