  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.
