The problem asks me to explain the correspondence between a subgraph of a hypercube and a block code.
I don't know how I'm supposed to use the fact that vertices are adjacent iff the vectors representing a vertex differ in exactly one position.
Please help me find a starting point.
Here's what immediately came to mind when I read your question. Hopefully it provides a starting point, as that's a rather vague question.
If we assume the block code is in binary for now, codewords look like some string of 0's and 1's. All we have to do is map it to a vertex of the hypercube by putting commas in between the 0's and 1's. For example:
Codeword $00110$ -> Point $(0,0,1,1,0)$
Of course, there may be less codewords than there are points on the hypercube. This is perhaps why the question refers to the subgraph of a hypercube instead of the whole hypercube. Also it may be important to note what happens if codewords are different lenghts. For example, which dimension of hypercube do you use if you have codewords $001$ and $0101$? If it's instantaneous, this shouldn't be too hard to deal with (you can add $0$s on the end of the words that are too short or something), but in other cases it may not be so easy.
As for the fact that vertices are adjacent iff the vectors representing a vertex differ in exactly one position. They probably want you to mention that if two codewords differ in exactly one position, those codewords will map to adjacent matrices. I'm not sure if this is relevant, but that means the Hamming Distance is 1 and that your code can't even detect (let alone correct) a single error.