When talking about FEC (forward error correction) block codes, some literature uses matrix terminology and some talks about polynomials. I know that the same block code could be expressed with either a generator matrix, or a generator polynomial, but I don't know what the relationship between them is.
Unfortunately I don't know enough about Galois fields, in this case GF(2) to figure this out.
- If you know the generator matrix of a block code, how do you get its generator polynomial?
- If you know the generator polynomial, how do you get the generator matrix?
And finally,
- Why are these equivalent?
You're dealing with two types of block codes : "linear" codes which can be completely defined by a generator matrix, and "cyclic" codes which can be completely defined by a generator polynomial. Cyclic codes are a subset of linear codes; so if you know the generator ploy of a cyclic code you can derive a generator matrix from it very easily (the process is straight forward and is described in many textbooks so I'll skip that, someone else might have a direct reference or online resource). Going the other way is not always possible since there are codes that are linear but not cyclic so no such polynomial exists for these. For some codes in use : Reed-Solomon codes are cyclic (and therefor linear), almost all other codes (LDPC codes in Wifi,...) are linear but not cyclic.