Optimal Error Correcting Index Codes for Extended Index Coding Problems

Abstract

The classical single-sender index coding problem (ICP) with error correction consists of a set of receivers, each receiving some transmissions erroneously. Each receiver demands a single message and has a subset of other messages as side information. The sender transmits an error correcting index code (ECIC) availing the knowledge of side information and demands of all the receivers, such that they are able to decode their demands correctly, even with at most δ erroneously received code symbols. An ICP ${{\mathcal{I}}_e}$ is called an extended ICP (EICP) of another ICP ${\mathcal{I}}$, if the fitting matrix of ${\mathcal{I}}$ is a submatrix of that of ${{\mathcal{I}}_e}$. The ICP ${\mathcal{I}}$ is said to be a sub-problem of ${{\mathcal{I}}_e}$. We construct optimal linear ECICs for some classes of EICPs. We first establish a parameter called the generalized independence number for a special class of EICPs, in terms of those of its sub-problems. Using this result, we construct optimal linear ECICs for a special subclass of EICPs. We identify some classes of EICPs, where optimal linear ECICs can be constructed using linear ECICs of the sub-problems, even when some of the ECICs of the sub-problems are sub-optimal. This is the first work according to the authors’ knowledge, where optimal linear ECICs for some classes of EICPs are constructed using those of the sub-problems.