  This paper focuses on error-correcting codes that can handle a predefined set
of specific error patterns. The need for such codes arises in many settings of
practical interest, including wireless communication and flash memory systems.
In many such settings, a smaller field size is achievable than that offered by
MDS and other standard codes. We establish a connection between the minimum
alphabet size for this generalized setting and the combinatorial properties of
a hypergraph that represents the prespecified collection of error patterns. We
also show a connection between error and erasure correcting codes in this
specialized setting. This allows us to establish bounds on the minimum alphabet
size and show an advantage of non-linear codes over linear codes in a
generalized setting. We also consider a variation of the problem which allows a
small probability of decoding error and relate it to an approximate version of
hypergraph coloring.
