In this paper, we study the codes $\mathcal C_k(n,q)$ arising from the incidence of points and $k$-spaces in $\text{PG}(n,q)$ over the field $\mathbb F_p$, with $q = p^h$, $p$ prime. We classify all codewords of minimum weight of the dual code $\mathcal C_k(n,q)^\perp$ in case $q \in \{4,8\}$. This is equivalent to classifying the smallest sets of even type in $\text{PG}(n,q)$ for $q \in \{4,8\}$. We also provide shorter proofs for some already known results, namely of the best known lower bound on the minimum weight of $\mathcal C_k(n,q)^\perp$ for general values of $q$, and of the classification of all codewords of $\mathcal C_{n-1}(n,q)$ of weight up to $2q^{n-1}$.