## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 163-185 |

Number of pages | 23 |

Journal | Combinatorica |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2008 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics