I have a generator matrix below: \begin{pmatrix}1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1\end{pmatrix}
$V$ is the binary linear code given by this matrix. Length $n = 5$, and dimension $K = 3$. I'm having trouble figuring out the process for finding all the code vectors from this matrix. I want to find them so I can find the min. distance of $V$. I should have $2^k$ codewords I think, so $2^3 = 8$ codewords total. What's the correct way for getting these codes?
Edit: So there are $2^k$ codewords, but I'm not sure how I multiply them together if the $\#$ of columns for the matrix is different than the number of rows for the vectors? IE matrix is $3\times6$ but the vectors are $3$ by $1$.
If you have a $k \times n$ generator matrix $G$ and message vector $m$ of length $k$, you can encode the message (i.e., find the code vector corresponding to that message vector) by computing $mG$.
To find all code vectors, simply repeat this computation for every possible value of $m$. As you observed, there are $2^k$ possible values of $m$ if you're working in binary.