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Let $C_{n-1}(n,q)$ be the code arising from the incidence of points andhyperplanes in the Desarguesian projective space PG($n,q$). Recently, Polverinoand Zullo proved that within this code, all non-zero code words of weight atmost $2q^{n-1}$ are scalar multiples of either the incidence vector of onehyperplane, or the difference of the incidence vectors of two distincthyperplanes. 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 incidencevectors of hyperplanes through a fixed $(n-3)$-space. Depending on the omittedvalue for $q$, we can lower the bound on the weight of $c$ to obtain the sameresults.

Small weight code words arising from the incidence of points and hyperplanes in PG($\boldsymbol{n,q}$)