Linear Network Coding, Linear Index Coding and Representable Discrete Polymatroids

TL;DR

This paper establishes a deep connection between linear network coding, index coding, and representable discrete polymatroids, providing new algorithms and characterizations for solutions over finite fields.

Contribution

It introduces the concept of discrete polymatroidal networks, linking their representability to linear fractional solutions, and generalizes existing matroid-based results to discrete polymatroids.

Findings

Linear fractional solutions exist if and only if the network is discrete polymatroidal and representable.

A construction algorithm for networks from certain discrete polymatroids is provided.

Representation of a discrete polymatroid corresponds to a linear fractional solution for the network.

Abstract

Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding and representable discrete polymatroids. We consider vector linear solutions of networks over a field $\mathbb{F}_q,$ with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. We define a \textit{discrete polymatroidal} network and show that a linear fractional solution over a field $\mathbb{F}_q,$ exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over $\mathbb{F}_q.$ An algorithm to construct networks starting from certain class of discrete polymatroids is provided. Every representation over $\mathbb{F}_q$ for the discrete polymatroid, results in a linear fractional solution over $\mathbb{F}_q$ for the constructed network. Next, we consider the index coding problem and show that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions which are determined by the index coding problem considered. El Rouayheb et. al. showed that the problem of finding a multi-linear representation for a matroid can be reduced to finding a \textit{perfect linear index coding solution} for an index coding problem obtained from that matroid. We generalize the result of El Rouayheb et. al. by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid.