How do I find a (7, 3) parity check code generator matrix?

Question

Our teacher didn't mention what polynomial we're to use, or anything else much. All he said was to read about it online, which I did and apparently I need some polynomial of degree four to find the matrix. How do you find this polynomial and why doesn't any polynomial work? Also, what do you do after you have the polynomial, like how do you choose what message to encode and such? Thanks for your help.

Answer 1

You're not giving us much to go on, but maybe the matrix $$A=\pmatrix{1&0&0&1&1&0&1\cr0&1&0&1&0&1&1\cr0&0&1&0&1&1&1\cr}$$ is the matrix you are looking for. If you want to encode a message, first you turn that message into a bunch of length 3 bitstrings (by any method you like), then you multiply each bitstring, $v$, by the matrix $A$, giving you a length 7 bitstring that you send out as the encoded message. Maybe you want to search for "Hamming code" for a fuller explanation.