Small weight code words arising from the incidence of points and hyperplanes in PG(n,q)

Let Cn−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 (J Comb Theory Ser A 158:1–11, 2018) proved that within this code, all non-zero code words of weight at most 2qn−1 are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We prove that all code words of weight at most (4q−O(q√))qn−2 are linear combinations of incidence vectors of hyperplanes through a common (n−3)-space. This extends previous results for large values of q.