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Do you have all $k^n$ strings of length $n$ using your $k$ symbol alphabet or a selection such as all consecutive 30 letter strings found in the DNA of some individual. The former is a Hamming graph but the latter is "subgraph of a Hamming graph" and I'm not sure how easy it is to say if a given graph can be realized in this way. Certainly given an edge the vertices adjacent to both ends are all adjacent to each other. So one could call any maximal clique a "line" and two points determine at most one line. One can see how to define "planes" etc.