Small Weight Code Words of Projective Geometric Codes

Abstract

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 $0\le j<k<n$. Firstly, we prove that all code words of ${\mathcal{C}}_{j,k}(n,q)$ up to weight $\left(3-\mathcal{O}\left(\frac{1}{q}\right)\right){\left[\genfrac{}{}{0ex}{}{k+1}{j+1}\right]}_q$ 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.

26 pages, 1 figure

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