Imagine I have $C (n,k,d)$ - linear code above $GF(q)$. Consider I know all words of this code.
$c_1, ..., c_M$ -- vectors from it.
If matrix H -- checking matrix (so any word from code will give zero in this way: $vH=0$ for this code will be
$H = [ c_1^t,...,c_M^t ] $
How can I find minimum distance in this code with such $ H$
Suppose you have a $k\times n$ generator matrix $G$ of the code. Then by elementary row operations and column permutations (which means you consider an equivalent code with the same minimum distance), the matrix has the form $G= (I\mid A)$, where $I$ is the $k\times k$ identity matrix. Then the $(n-k)\times n$ matrix $H = (-A^T\mid I)$ is a check matrix, since $GH^T=0$. Now proceed to find the minimum number of linearly independent columns of $H$. If this number is $d-1$, the minimum distance is $\geq d$ (with equality if the code has a word of Hamming weight $d$).