Small weight codewords of projective geometric codes II

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, \(j < k\).