  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
implicit bound on the length of an optimal linear ECIC.
  Further, the problem of error-correcting decoding by a linear ECIC is
studied. It is shown that in order to decode correctly the desired symbol, the
decoder is required to find one of the vectors, belonging to an affine space
containing the actual error vector. The syndrome decoding is shown to produce
the correct output if the weight of the error pattern is less or equal to the
error-correcting capability of the corresponding ECIC.
  Finally, the notion of static ECIC, which is suitable for use with a family
of instances of an index coding problem, is introduced.
