如何使用奇偶校验矩阵进行编码？

Question

I want to encode information block with parity check matrix.

I have the hamming code (7,4,3) with parity check matrix H and I want to encode the information block m = [0 0 1] .

H = [1 1 0 1 1 0 0
     1 0 1 1 0 1 0
     0 1 1 1 0 0 1];

With generator matrix I just use this formula : codeword = mod(word*G,2) , but I have no idea how to encode using the parity check matrix.

Thanks!

Answer 1

As mentioned here , For encoding you should use generator matrix (G) not parity check matrix (as its name shows this!). So, encoded vector would be G*m' .

Answer 2

As stated in the previous answer, you need to get the generator matrix in order to encode. It is straightforward to transform between a generator and parity matrix if they are in standard form, which requires the code to be systematic in the first k positions (see [1]).

If the parity matrix is not in that form, you may use elementary row operations to put it into standard form, and then obtain the generator matrix (see [4]). This changes the code to a different, but "equivalent" code (see [5]).

My cursory search found no other way to find a generator matrix from an arbitrary parity matrix. I suspect it is difficult, impractical, or impossible to do so, but I do not have a definitive answer.

REFERENCES AND DISCUSSION
Note that these references are not necessarily authoritative; but I find them to be reasonable and plausible.

[1] https://en.wikipedia.org/wiki/Parity-check_matrix For an (n,k) block code, the standard form for a generator matrix is G = [ I(k) P ]. G is a (k,n) matrix. I(k) is the (k,k) identity matrix. P is a (k,nk) matrix.

The parity matrix is then H = [ -P^TI(nk) ]. H is the (nk,n) parity matrix, corresponding to G. -P^T is the negative transposition of P. You can leave out the negation for binary codes. I(nk) is the (nk, nk) identity matrix.

So, if you have a parity matrix in the form I have shown, the generator matrix is G = [ I(k) P ].

[2] Pless, Vera (1998), Introduction to the Theory of Error-Correcting Codes (3rd ed.), Wiley Interscience, ISBN 0-471-19047-0. This is cited by [1].

[3] https://www.mathworks.com/help/comm/ref/gen2par.html Matlab provides a function for converting between the generator/parity (gen2par), but it seems to require the standard form. The function seems to rely on this standard form to detect which one you have provided, and then change it into the other.

[4] https://math.stackexchange.com/questions/1490627/finding-generator-matrix-for-binary-linear-code-given-parity-check-matrix This reference proposes using elementary row operations to transform a parity matrix into standard form, then transforming into a generator matrix.

[5] https://math.stackexchange.com/questions/1684808/row-column-operations-of-a-parity-check-generator-matrix-for-a-linear-code This reference explains that elementary row operations on a parity or generator matrix will change the code, because the coded representation of a particular set of inputs will change, but the code may be considered "equivalent" because the set of codewords (hence distance properties) are unchanged.