Small weight codewords of projective geometric codes II

The p-ary linear code C_kn,q is defined as the row space of the incidence matrix A of k-spaces and points of PGn,q. It is known that if q is square, a codeword of weight q^kq+Oq^k-1 exists that cannot be written as a linear combination of at most 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⩾32 is a composite prime power, every codeword of C_kn,q up to weight Oq^kq is a linear combination of at most q rows of A. We also generalise this result to the codes C_j,kn,q, which are defined as the p-ary row span of the incidence matrix of k-spaces and j-spaces, j<k.