The minimum weight of the code of intersecting lines in $\mathrm{P}\mathrm{G}(3,q)$

We characterise the minimum weight codewords of the $p$-ary linear code of intersecting lines in $\mathrm{P}\mathrm{G}(3,q)$, $q=p^h$, $q\ge 19$, $p$ prime, $h\ge 1$. If $q$ is even, the minimum weight equals $q^3+q^2+q+1$. If $q$ is odd, the minimum weight equals $q^3+2q^2+q+1$. For $q$ even, we also characterise the codewords of second smallest weight.