Parity check matrices and product representations of squares

Let N_F (n,k,r) denote the maximum number of columns in an n-row matrix with entries in a finite field N_F in which each column has at most r nonzero entries and every k columns are linearly independent over N _F. We obtain near-optimal upper bounds for N_F (n,k,r) in the case k > r. Namely, we show that N_F (n,k,r) ≫ n ^{r/2 + cr/k} where c ≈ 4/3 for large k. Our method is based on a novel reduction of the problem to the extremal problem for cycles in graphs, and yields a fast algorithm for finding short linear dependencies. We present additional applications of this method to a problem on hypergraphs and a problem in combinatorial number theory.