A note on small weight codewords of projective geometric codes and on the smallest sets of even type

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 $2q^{n-1}$.