Matroidal Entropy Functions: Constructions, Characterizations and Representations

TL;DR

This paper explores matroidal entropy functions, providing new characterizations and representations for functions induced by regular matroids and specific matroids with particular p-characteristics, impacting information theory applications.

Contribution

It introduces constructions and characterizations of matroidal entropy functions, especially for regular matroids and those sharing p-characteristics with uniform matroids.

Findings

Characterized matroidal entropy functions for regular matroids.

Identified matroids with the same p-characteristic set as U_{2,4}.

Developed variable strength arrays for matroid operations.

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 $U_{2,4}$.