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. Although there are restrictions.
Define a "line" to be any maximal clque (set of adjacent points) then 2 points are on at most one line. Then lines have size $k$ or less. Planes are less clear to me.