On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual

In this paper, we study the $p$-ary linear code $C(PG(n,q))$, $q=p^h$, $p$ prime, $h\geq 1$, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space $PG(n,q)$, and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in $PG(n,q)$ and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of $C(PG(n,q))$ and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in $PG(2,q)$. We present an improved upper bound on this minimum weight and show that we can drop the divisibility condition on the weight of the codewords in Sachar's lower bound \cite{sachar}.