Vector linear error correcting index codes and discrete polymatroids

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

I. Introduction

The index coding problem introduced by Birk and Kol [1] involves a source which generates a set of messages and set of receivers which demand messages. Each receiver has prior knowledge of a portion of the message called side- The source uses the side-information available at all the receivers to find a transmission scheme of minimum number of transmissions, which satisfies all the demands of the receivers. Bar-Yossef et al. [2] studied the index coding problem and found that the length of the optimal linear index code is equal to the minrank of a related graph. Lubetzky and Stay [3] showed that non-linear scalar codes are better than linear scalar ones. The connection between multi-linear representation of matroids and index coding was studied by El Rouayheb, Sprintson and Georghiades [4]. It was shown in [5] that a vector 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.