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A Maximum Distance Separable code over an alphabet $F$ is defined via anencoding 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 givenparameters $n$ and $k$ is unknown and forms one of the central open problems incoding theory. The paper initiates the study of the alphabet size of codes in ageneralized setting where the coding scheme is required to handle apre-specified subset of all possible erasure patterns, naturally represented byan $n$-vertex $k$-uniform hypergraph. We relate the minimum possible alphabetsize of such codes to the strong chromatic number of the hypergraph and analyzethe tightness of the obtained bounds for both the linear and non-linearsettings. We further consider variations of the problem which allow a smallprobability of decoding error.

Minimizing the alphabet size of erasure codes with restricted decoding sets