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[Answers] About codes over $\mathbb{F}_2$

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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?

Asked By : user6818

Answered By : Yuval Filmus

  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 $\mathbb{F}_2$.

  3. There are several NP-complete problems related to codes; Madhu Sudan has an entire lecture on these. One of them is, given a generator matrix for a linear code, determine the minimum weight of a non-zero codeword (i.e., the minimum distance).

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