I suspect it is difficult, impractical, or impossible to do so, but I do not have a definitive answer. My cursory search found no other way to find a generator matrix from an arbitrary parity matrix. This changes the code to a different, but "equivalent" code (see ). 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 ). 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 ). As stated in the previous answer, you need to get the generator matrix in order to encode.
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