I have an extremely basic question about Parity Check Matrix. Let's call it H and consider a codeword: I have been told that through H we may verify if it is a real codeword or not (and consider the case it is).
Precisely, if the product c*H' = 0 (where ' denotes transpose), it is a true codeword (I read this sentence here, on slide 5).
But in some situations I computed that quantity for a true codeword and I did not get a sequence of 0s (and sometimes I got a vector made of numbers which are not 0 or 1, but as 2, 4 etc).
Example
Consider Hamming Code 7,4 with parity check and generator matrices equal to, respectively,
If you consider an input sequence b = [1 0 1] and you find the corresponding codeword, you get:
c = b*G = [1 0 1 0 0 1 1]
If you multiply this by H' you get [2 2 2 2], which is not [0 0 0 0].
Hamming code is a binary code, all arithmetic is modulo 2, so what you have is the zero vector.