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