An empty interval in the spectrum of small weight codewords in the code from points and k-spaces of PG(n, q)

Let $\C(n,q)$ be the $p$-ary linear code defined by the incidence matrix of points and $k$-spaces in $\PG(n,q)$, $q=p^h$, $p$ prime, $h\geq 1$. In this paper, we show that there are no codewords of weight in $]\frac{q^{k+1}-1}{q-1},2q^k[$ in $\C(n,q)\setminus\mathrm{C}_{n-k}(n,q)^\bot$ which implies that there are no codewords with this weight in $\C(n,q)\setminus \C(n,q)^{\bot}$ if $k\geq n/2$. In particular, for the code $\mathrm{C}_{n-1}(n,q)$ of points and hyperplanes of $\PG(n,q)$, we exclude all codewords in $\mathrm{C}_{n-1}(n,q)$ with weight in $]\frac{q^n-1}{q-1},2q^{n-1}[$. This latter result implies a sharp bound on the weight of small weight codewords of $\mathrm{C}_{n-1}(n,q)$, a result which was previously only known for general dimension for $q$ prime and $q=p^2$, with $p$ prime, $p>11$, and in the case $n=2$, for $q=p^3$, $p\geq 7$.