On the Index Coding Problem and its Relation to Network Coding and Matroid Theory

TL;DR

This paper explores the index coding problem, demonstrating its deep connections to network coding and matroid theory, and shows how vector linear codes outperform scalar codes but are still insufficient for optimal solutions.

Contribution

It establishes reductions from network coding and matroid problems to index coding, revealing the limitations of linear codes and advancing theoretical understanding.

Findings

Vector linear codes outperform scalar linear codes.

Linear codes are insufficient for optimal index coding solutions.

Reductions connect index coding with network coding and matroid theory.

Abstract

The \emph{index coding} problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless ad-hoc networks. An instance of the index coding problem includes a sender that holds a set of information messages $X=\{x_1,...,x_k\}$ and a set of receivers $R$. Each receiver $\rho=(x,H)\in R$ needs to obtain a message $x\in X$ and has prior \emph{side information} comprising a subset $H$ of $X$. The sender uses a noiseless communication channel to broadcast encoding of messages in $X$ to all clients. The objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the receivers' demands with \emph{zero error}. In this paper, we analyze the relation between the index coding problem, the more general network coding problem and the problem of finding a linear representation of a matroid. In particular, we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem. Our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem. Specifically, we show that \emph{vector linear codes} outperform scalar linear codes and that vector linear codes are insufficient for achieving the optimum number of transmissions.