### Abstract

Let N_{double-struck F sign}(n, k, r) denote the maximum number of columns in an n-row matrix with entries in a finite field double-struck F sign in which each column has at most r nonzero entries and every k columns are linearly independent over double-struck F sign. Such sparse parity check matrices are fundamental tools in coding theory, derandomization and complexity theory. We obtain near-optimal theoretical upper bounds for N _{double-struck F sign}(n, k, r) in the important case k > r, i.e. when the number of correctible errors is greater than the weight. Namely, we show that N_{double-struck F sign}(n, k, r) = O(n^{r/2+4r/3k}). The best known (probabilistic) lower bound is N _{double-struck F sign}(n, k, r) = Ω(n^{r/2+r/2k-2}), while the best known upper bound in the case k > r was for k a power of 2, in which case N_{double-struck F sign}(n, k, r) = Ω(n ^{r/2+1/2}). 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 dependences in large sets of sparse vectors. In the full version of this paper we present additional applications of this method to problems in combinatorial number theory.

Original language | English (US) |
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Title of host publication | Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05 |

Pages | 1749-1752 |

Number of pages | 4 |

DOIs | |

State | Published - Dec 1 2005 |

Externally published | Yes |

Event | 2005 IEEE International Symposium on Information Theory, ISIT 05 - Adelaide, Australia Duration: Sep 4 2005 → Sep 9 2005 |

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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Volume | 2005 |

ISSN (Print) | 2157-8099 |

### Other

Other | 2005 IEEE International Symposium on Information Theory, ISIT 05 |
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Country | Australia |

City | Adelaide |

Period | 9/4/05 → 9/9/05 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Information Systems
- Modeling and Simulation
- Applied Mathematics

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## Cite this

*Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05*(pp. 1749-1752). [1523645] (IEEE International Symposium on Information Theory - Proceedings; Vol. 2005). https://doi.org/10.1109/ISIT.2005.1523645