Small weight codewords of projective geometric codes II

Abstract

The $p$-ary linear code $\mathcal {C}_{k}\!\left( n,q\right)$ is defined as the row space of the incidence matrix $A$ of $k$-spaces and points of $\textrm{PG}\!\left( n,q\right)$. 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 $\mathcal {C}_{k}\!\left( n,q\right)$ 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 $\mathcal {C}_{j,k}\!\left( n,q\right)$, which are defined as the $p$-ary row span of the incidence matrix of k-spaces and j-spaces, $j < k$.