I was looking through these notes but I am not sure I can locate the answer to these questions of mine - it would be great if someone can just even point out what to look for!
So any set of binary vectors can be seen as "code"?
Let $M$ be a $d\times m$ matrix over $\mathbb{F}_2$ and let $X(M)$ be the graph on the binary vectors of length $d$, where two vectors are adjacent if their difference is a column of $M$. (does this mean that before comparing when one is taking a bit-wise difference of the vectors one is equating $-1$ to $1$ as would be inside $\mathbb{F}_2$?)
Does this above construction have a name? Any motivations?
What is this NP-hard question about finding a "minimum weight code" among a set of binary vectors? Can someone kindly give the precise definition?