Parity check matrices and product representations of squares

Abstract

Let $$ N_{\mathbb{F}} $$ (n,k,r) denote the maximum number of columns in an n-row matrix with entries in a finite field $$ \mathbb{F} $$ in which each column has at most r nonzero entries and every k columns are linearly independent over $$ \mathbb{F} $$ . We obtain near-optimal upper bounds for $$ N_{\mathbb{F}} $$ (n,k,r) in the case k > r. Namely, we show that $$ N_\mathbb{F} (n,k,r) \ll n^{\frac{r} {2} + \frac{{cr}} {k}} $$ where $$ c \approx \frac{4} {3} $$ 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.