Small weight codewords of projective geometric codes II.

The p -ary linear code C k n , q is defined as the row space of the incidence matrix A of k -spaces and points of PG n , q . It is known that if q is square, a codeword of weight q k q + O q 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 k n , q up to weight O q k q is a linear combination of at most q rows of A . We also generalise this result to the codes C j , k n , q , which are defined as the p -ary row span of the incidence matrix of k-spaces and j-spaces, j < k . [ABSTRACT FROM AUTHOR]