Index Coding and Error Correction

TL;DR

This paper extends index coding to include error correction, establishing bounds on code length, proposing constructions for large alphabets, and analyzing decoding methods, thereby advancing the robustness of index coding schemes.

Contribution

It generalizes index coding to error correction, derives bounds, and proposes constructions, including for large alphabets, with analysis of decoding strategies.

Findings

Established Singleton, α, and κ bounds for error-correcting index codes.

Proposed concatenation-based construction attains the Singleton bound for large alphabets.

Analyzed syndrome decoding for error correction in index coding schemes.

Abstract

A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998). In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols $\delta$, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest-length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the $\alpha$-bound and the $\kappa$-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another inexplicit bound on the length of an optimal linear ECIC. Finally, the decoding of linear ECIC's is discussed. The syndrome decoding is shown to output the exact message if the weight of the error vector is less or equal to the error-correcting capability of the corresponding ECIC.