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Matroidal Entropy Functions: Constructions, Characterizations and Representations

Qi Chen, Minquan Cheng, Baoming Bai

TL;DR

This work defines matroidal entropy functions as $\mathbf{h} = \log v \cdot \mathbf{r}_M$ and develops a VOA-based framework that links matroids, partitions, and information-theoretic constraints. It provides constructions for VOAs under deletions, contractions, and binary matroid operations, and uses these to characterize matroidal entropy for regular matroids and for matroids sharing the $p$-characteristic set with $U_{2,4}$. A key contribution is the complete identification of the p-characteristic set for regular matroids and the characterization of a broad class of matroids with the same $\chi$ as $U_{2,4}$, via 3-connected components and wheel/whirl structures. The results yield both structural insights into the entropy region and practical implications for network coding and related information-theoretic problems, where matroidal entropy functions underpin capacity characterizations and code constructions.

Abstract

Matroidal entropy functions are entropy functions in the form $\mathbf{h} = \log v \cdot \mathbf{r}_M$ , where $v \ge 2$ is an integer and $\mathbf{r}_M$ is the rank function of a matroid $M$. They can be applied into capacity characterization and code construction of information theory problems such as network coding, secret sharing, index coding and locally repairable code. In this paper, by constructing the variable strength arrays of some matroid operations, we characterized matroidal entropy functions induced by regular matroids and some matroids with the same p-characteristic set as uniform matroid $U_{2,4}$.

Matroidal Entropy Functions: Constructions, Characterizations and Representations

TL;DR

This work defines matroidal entropy functions as and develops a VOA-based framework that links matroids, partitions, and information-theoretic constraints. It provides constructions for VOAs under deletions, contractions, and binary matroid operations, and uses these to characterize matroidal entropy for regular matroids and for matroids sharing the -characteristic set with . A key contribution is the complete identification of the p-characteristic set for regular matroids and the characterization of a broad class of matroids with the same as , via 3-connected components and wheel/whirl structures. The results yield both structural insights into the entropy region and practical implications for network coding and related information-theoretic problems, where matroidal entropy functions underpin capacity characterizations and code constructions.

Abstract

Matroidal entropy functions are entropy functions in the form , where is an integer and is the rank function of a matroid . They can be applied into capacity characterization and code construction of information theory problems such as network coding, secret sharing, index coding and locally repairable code. In this paper, by constructing the variable strength arrays of some matroid operations, we characterized matroidal entropy functions induced by regular matroids and some matroids with the same p-characteristic set as uniform matroid .
Paper Structure (18 sections, 23 theorems, 39 equations, 3 figures, 1 table)

This paper contains 18 sections, 23 theorems, 39 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matroids
  4. Variable strength orthogonal array
  5. Constructions
  6. Deletions, contractions and minors
  7. Series and parallel connections
  8. Direct sum and $2$-sum
  9. P-characteristic set of two classes of matroids
  10. Regular matroid
  11. Matroids with the same p-characteristic set as $U_{2,4}$
  12. Discussion
  13. Entropy functions on the extreme rays of polymatroidal region
  14. Connectivity
  15. Excluded minor and classification of matroids according to its p-characteristic set
  16. ...and 3 more sections

Key Result

Lemma 1

For a $\mathrm{VOA}(M,v)$$\mathbf{T}$ with matroid $M=(N,\mathbf{r})$ and integer $v\ge 2$,

Figures (3)

  • Figure 1: The graph whose cycle matroid is $M$ in Exmaple \ref{['asdf']}.
  • Figure 2: The wheel graph $\mathcal{W}_r$
  • Figure 3: The $(4, 2)$-combination network

Theorems & Definitions (62)

  • Definition 1: O11
  • Example 1
  • Example 2
  • Definition 2: HSS99
  • Definition 3: CCB21
  • Remark
  • Example 3
  • Lemma 1
  • Theorem 1
  • Remark
  • ...and 52 more