Vector Linear Error Correcting Index Codes and Discrete Polymatroids

TL;DR

This paper explores the relationship between vector linear differential error correcting index codes and discrete polymatroids, establishing conditions for their existence based on representability of these polymatroids.

Contribution

It introduces the concept of differential error correcting index codes and characterizes their existence through the framework of representable discrete polymatroids.

Findings

Existence of vector linear differential error correcting index codes is equivalent to certain conditions on representable discrete polymatroids.

Extends previous work on error correcting index codes to a more general scenario with receiver-specific error correction.

Connects index coding with discrete polymatroid theory, broadening the theoretical foundation.

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 multi linear 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. We show that vector linear differential error correcting index code exists if and only if there exists a representable discrete polymatroid satisfying certain conditions