Abstract. In this paper, we study the codes [Formula: see text] arising from the incidence of points and [Formula: see text]-spaces in [Formula: see text] over the field [Formula: see text], with [Formula: see text], [Formula: see text] prime. We classify all codewords of minimum weight of the dual code [Formula: see text] in case [Formula: see text]. This is equivalent to classifying the smallest sets of even type in [Formula: see text] for [Formula: see text]. We also provide shorter proofs for some already known results, namely, of the best known lower bound on the minimum weight of [Formula: see text] for general values of [Formula: see text], and of the classification of all codewords of [Formula: see text] of weight up to [Formula: see text].

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.