I am given the code $C[8,4,d]$ over $\mathbb{F}_3$ with generator polynomial $g(x)=x^4 + 1$. From theory I know that the check polynomial is given by the division of $x^8 - 1$ by $x^4+1$. This gives precisely $h(x)=x^4 + 2$.
From this, I can build the following parity check matrix $H$:
$\begin{bmatrix} 1 & 0 & 0 & 0 & 2 & 0 &0 &0 \\ 0 & 1 & 0 & 0 & 0 & 2 &0 &0 \\ 0 & 0 & 1 & 0 & 0 & 0 &2 &0 \\ 0 & 0 & 0 & 1 & 0 & 0 &0 &2 \end{bmatrix}$
My question is: WHAT IS THE MINIMUM DISTANCE?
Actually, if I sum all the columns I find the vector $(3,3,3)=(0,0,0)\mod3$. So, it seems to me that $8$ columns are linearly independent, and since every set of $7$ columns is linearly independent, I'd say that the minimum distance is 8.
Is everthing okay?