Four codewords of a linear $(7,3)$ code $C$ are given to me in a question with no other details, which are$$0110101,1001101,1001011,0000000.$$I am asked to write down the missing codewords.
I have tried choosing generator matrix randomly but could not find the correct codewords. However i was able to find the correct codewords by using linear code word property of xor of two codewords will also be a codeword.
Now I have all eight codewords and want to make a generator matrix or parity matrix from given eight codewords but I am confused on how to create it. I have several documents but they seem to be using parity matrix which I do not have here.
The first three words are linearly independent over ${\Bbb Z}_2$. This allows you to write down a $3\times 7$ generator matrix $G$ with the codewords as rows. Then you find the code as $C = \{aG\mid a\in{\Bbb Z}_2^3\}$.