Error Correcting Index Codes and Matroids

TL;DR

This paper explores the relationship between error correcting index codes and matroid theory, introducing differential error correction where each receiver can correct a different number of errors, and establishing conditions for scalar linear solutions.

Contribution

It generalizes existing error correcting index codes by allowing different error correction capabilities per receiver and links these codes to representable matroids.

Findings

Established a link between differential error correcting index codes and certain matroids.

Defined matroidal differential error correcting index codes.

Proved scalar linear differential error correcting index codes exist if and only if they are matroidal and associated with a representable matroid.

Abstract

The connection between index coding and matroid theory have been well studied in the recent past. El Rouayheb et al. established a connection between multilinear representation of matroids and wireless index coding. Muralidharan and Rajan showed that a vector linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions. Recently index coding with erroneous transmission was considered by Dau et al.. Error correcting index codes in which all receivers are able to correct a fixed number of errors was studied. In this paper we consider a more general scenario in which each receiver is able to correct a desired number of errors, calling such index codes differential error correcting index codes. A link between differential error correcting index codes and certain matroids is established. We define matroidal differential error correcting index codes and we show that a scalar linear differential error correcting index code exists if and only if it is matroidal differential error correcting index code associated with a representable matroid.