and parity matrix $H=\begin{bmatrix}P^T, I_{n-m}\end{bmatrix}$
I have a questions:
for example, let $C=\left ( 0001,1111 \right )$ so $d(C)=3$, how come $H^T$ is related to the parity matrix?According to the theorem, there should be $3-1=2$ rows of $H^T$ are linearly indenpendent and 3 rows are linearly denpendent. from my understanding, the $P$ part in the generator matrix $G$ can be anything, so $P^T$ could be anything. In this expample, $H^T=\begin{bmatrix} 1 & 1 &0 \\ 1 & 0 &1 \\ 0 & 1& 1\\ 1 & 1&1 \\ 1& 0 & 0\\ 0& 1& 0\\ 0& 0 & 1 \end{bmatrix}$, but we can see that the last 3 rows are linearly indenpendent which contradicts the theorem