Let $C$ be the ternary linear code with parity check matrix
$$H=\begin{pmatrix} 2 & 0 & 1 & 1 & 1 \\ 0 & 2 & 2 & 0 & 0 \\ 0 & 0 & 2 & 2 & 1 \\ 1 & 1 & 2 & 2 & 0 \end{pmatrix}.$$ What is the minimum distance of $C$, denoted $d(C)$?
From my understanding, it has to do with the linear dependent columns of the matrix, but I do not know how to solve this problems.
Hint: The code is given by $C=\{ vH \mid v\in{\Bbb F}_3^4\}\subseteq {\Bbb F}_3^5$, where all vectors are row vectors.
Consider the nonzero linear combinations of the columns of $H$.
Two different columns are not linearly independent (in this case one column would be the $\Bbb F_3$-multiple of another). Thus the minimum distance cannot be $2$.
Next consider three different columns of $H$. If you can linearly combine three columns to $0$, the minimum distance will be $3$.
And so on.
The smallest number of such linear combinations is the minimum distance of $C$.