Let N-F(n, k, r) denote the maximum number of columns in an n-row matrix with entries in a finite field F in which each column has at most r nonzero entries and every k columns are linearly independent over IF. We obtain near-optimal upper bounds for NF (n, k,r) in the case k > r. Namely, we show that N-F(n, k, r) n(r/2+cr/k) where c approximate to 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.
展开▼
