Codes arising from incidence matrices of points and hyperplanes in PG(n,q)

In this paper we completely characterize the words with second minimum weight in the p-ary linear code generated by the rows of the incidence matrix of points and hyperplanes of PG(n,q), with q=ph and p prime, proving that they are the scalar multiples of the difference of the incidence vectors of two distinct hyperplanes of PG(n,q).