Hamming (7,4) and Parity Matrix

Question

currently I am studying a few introductory things on Coding Theory but I have a small problem on understanding a few things.

I really cant understand how can I create a Parity matrix for H(7,4) if I don't have the Generator.

I saw some implementations on Matlab for example, and they use something called parity submatrix, and from this they create the H and the G.

For example, here is defined a parity-sub matrix (3 columns and 4 rows) for a Hamming(7,4) code: A = [ 1 1 1;1 1 0;1 0 1;0 1 1 ]; The author of this states: "Parity submatrix-Need binary(decimal combination of 7,6,5,3)"

Now the real question is, are there some specific rules on how to create the parity sub matrix or no?

Thanks in advance!

Answer 1

The parity-check matrix of a $[2^n-1, 2^n-1-n]$ Hamming code is of size $n\times (2^n-1)$ and its columns are the $2^n-1$ distinct nonzero binary $n$-tuples or vectors. If you choose the rightmost $n$ columns as the $n\times n$ identity matrix, you get what is called a systematic code in which the first $2^n-1-n$ symbols in each codeword are the information symbols and the last $n$ symbols are the parity-check symbols. For the case $n=3$ considered here, the last $3$ columns are the base-$2$ representations of the integers $1, 2, 4$, and so the first four columns must be the base-$2$ representations of the other integers in the range $[1, 7]$ viz., $7,6,5,3$ which is what is in your question. Note that ordering of these four $3$-tuples in the parity-check matrix is a matter of choice.