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

In this paper, we study the $p$-ary linear code $C_{k}(n,q)$, $q=p^h$, $p$ prime, $h\geq 1$, generated by the incidence matrix of points and $k$-dimensional spaces in $PG(n,q)$. For $k\geq n/2$, we link codewords of $C_{k}(n,q)\setminus C_{k}(n,q)^\perp$ of weight smaller than $2q^k$ to $k$-blocking sets. We first prove that such a $k$-blocking set is uniquely reducible to a minimal $k$-blocking set, and exclude all codewords arising from small linear $k$-blocking sets. For $k<n/2$, we present counterexamples to lemmas valid for $k\geq n/2$. Next, we study the dual code of $C_k(n,q)$ and present a lower bound on the weight of the codewords, hence extending the results of Sachar \cite{sachar} to general dimension.