Small Weight Code Words of Projective Geometric Codes

We investigate small weight code words of the $p$-ary linear code $\mathcal C_{j,k}(n,q)$ generated by the incidence matrix of $k$-spaces and $j$-spaces of PG$(n,q)$ and its dual, with $q$ a prime power and $0 \leq j < k < n$. Firstly, we prove that all code words of $\mathcal C_{j,k}(n,q)$ up to weight $\left(3 - \mathcal{O}\left(\frac 1 q \right) \right) \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$ are linear combinations of at most two $k$-spaces (i.e. two rows of the incidence matrix). As for the dual code $\mathcal C_{j,k}(n,q)^\perp$, we manage to reduce both problems of determining its minimum weight (1) and characterising its minimum weight code words (2) to the case $\mathcal C_{0,1}(n,q)^\perp$. This implies the solution to both problem (1) and (2) if $q$ is prime and the solution to problem (1) if $q$ is even.