  The $p$-ary linear code $\mathcal C_{k}(n,q)$ is defined as the row space of
the incidence matrix $A$ of $k$-spaces and points of $\text{PG}(n,q)$. It is
known that if $q$ is square, a codeword of weight $q^k\sqrt{q}+\mathcal O
\left( q^{k-1} \right) $ exists that cannot be written as a linear combination
of at most $\sqrt{q}$ rows of $A$. Over the past few decades, researchers have
put a lot of effort towards proving that any codeword of smaller weight does
meet this property. We show that if $ q \geqslant 32 $ is a composite prime
power, every codeword of $\mathcal C_k(n,q)$ up to weight $\mathcal O \left(
{q^k\sqrt{q}} \right) $ is a linear combination of at most $\sqrt{q}$ rows of
$A$. We also generalise this result to the codes $\mathcal C_{j,k}(n,q) $,
which are defined as the $p$-ary row span of the incidence matrix of $k$-spaces
and $j$-spaces, $j < k$.
