The problem does not say whether I am allowed to expand or not, so I am guessing that I need to find an answer by any means necessary.It is not clear to me can you go the expansion field or not.

If not, the only code I see here is Reed-Solomon [6,4,3]:

in GF(7), primitive element is 5, so single error-correcting code with d=3 has generator: (x - 5)(x - 4).

To reach requested code length you have to expand code. Do you allow to do this?

Otherwise, just go to expansion GF(49) and then shorten the big code.

So I would have the identity matrix (6x6) for part of my generator matrix (9x6).

(I'm trying to derive the generator matrix... then it will be simple for me to convert it to parity check matrix.)

Any further help regarding the expand route would be greatly appreciated.