  Over the past few years, the codes $\mathcal{C}_{n-1}(n,q)$ arising from the
incidence of points and hyperplanes in projective spaces $\text{PG}(n,q)$
attracted lots of attention. In particular, small weight codewords of
$\mathcal{C}_{n-1}(n,q)$ were investigated. In light of this, our main result
states that, if $q$ is large enough and not prime, a codeword having weight
smaller than roughly $\frac{1}{2^{n-2}}q^{n-1}\sqrt{q}$ is a linear combination
of a few hyperplanes. Consequently, we use this result to provide a
graph-theoretical description of minimal codewords in $\mathcal{C}_{n-1}(n,q)$.
