  A Maximum Distance Separable code over an alphabet $F$ is defined via an
encoding function $C:F^k \rightarrow F^n$ that allows to retrieve a message $m
\in F^k$ from the codeword $C(m)$ even after erasing any $n-k$ of its symbols.
The minimum possible alphabet size of general (non-linear) MDS codes for given
parameters $n$ and $k$ is unknown and forms one of the central open problems in
coding theory. The paper initiates the study of the alphabet size of codes in a
generalized setting where the coding scheme is required to handle a
pre-specified subset of all possible erasure patterns, naturally represented by
an $n$-vertex $k$-uniform hypergraph. We relate the minimum possible alphabet
size of such codes to the strong chromatic number of the hypergraph and analyze
the tightness of the obtained bounds for both the linear and non-linear
settings. We further consider variations of the problem which allow a small
probability of decoding error.
