  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 $q=p^h$ and $p$ prime, proving that
they are the scalar multiples of the difference of the incidence vectors of two
distinct hyperplanes of $PG(n,q)$.
