  Let $C_{n-1}(n,q)$ be the code arising from the incidence of points and
hyperplanes in the Desarguesian projective space PG($n,q$). Recently, Polverino
and Zullo proved that within this code, all non-zero code words of weight at
most $2q^{n-1}$ are scalar multiples of either the incidence vector of one
hyperplane, or the difference of the incidence vectors of two distinct
hyperplanes. We improve this result, proving that when $q>17$ and
$q\notin\{25,27,29,31,32,49,121\}$, all code words of weight at most
$(4q-\sqrt{8q}-\frac{33}{2})q^{n-2}$ are linear combinations of incidence
vectors of hyperplanes through a fixed $(n-3)$-space. Depending on the omitted
value for $q$, we can lower the bound on the weight of $c$ to obtain the same
results.
