# About codes over $\mathbb{F}_2$

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?

• Please post only one question per post. – Raphael Apr 9 '15 at 20:44

1. A binary code is a set of vectors in $\mathbb{F}_2^n$ for some $n$.
2. Presumably the context in which you encountered this construction is a motivation for it. It's a particular case of a more general construction known as a Cayley graph, though perhaps this particular case has a specific name. You are right that all arithmetic is done in $\mathbb{F}_2$.