### 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 |

Volume | 2005 |

DOIs | |

State | Published - 2005 |

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

### Other

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

Country | Australia |

City | Adelaide |

Period | 9/4/05 → 9/9/05 |

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### ASJC Scopus subject areas

- Electrical and Electronic Engineering

### Cite this

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

**Improved bounds on the size of sparse parity check matrices.** / Naor, Assaf; Verstraete, Jacques.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05.*vol. 2005, 1523645, pp. 1749-1752, 2005 IEEE International Symposium on Information Theory, ISIT 05, Adelaide, Australia, 9/4/05. https://doi.org/10.1109/ISIT.2005.1523645

}

TY - GEN

T1 - Improved bounds on the size of sparse parity check matrices

AU - Naor, Assaf

AU - Verstraete, Jacques

PY - 2005

Y1 - 2005

N2 - Let Ndouble-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 Ndouble-struck F sign(n, k, r) = O(nr/2+4r/3k). The best known (probabilistic) lower bound is N double-struck F sign(n, k, r) = Ω(nr/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 Ndouble-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.

AB - Let Ndouble-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 Ndouble-struck F sign(n, k, r) = O(nr/2+4r/3k). The best known (probabilistic) lower bound is N double-struck F sign(n, k, r) = Ω(nr/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 Ndouble-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.

UR - http://www.scopus.com/inward/record.url?scp=33749425389&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749425389&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2005.1523645

DO - 10.1109/ISIT.2005.1523645

M3 - Conference contribution

SN - 0780391519

SN - 9780780391512

VL - 2005

SP - 1749

EP - 1752

BT - Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05

ER -