Improved bounds on the size of sparse parity check matrices

Assaf Naor, Jacques Verstraete

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

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.

Original languageEnglish (US)
Title of host publicationProceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05
Pages1749-1752
Number of pages4
Volume2005
DOIs
StatePublished - 2005
Event2005 IEEE International Symposium on Information Theory, ISIT 05 - Adelaide, Australia
Duration: Sep 4 2005Sep 9 2005

Other

Other2005 IEEE International Symposium on Information Theory, ISIT 05
CountryAustralia
CityAdelaide
Period9/4/059/9/05

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Number theory

ASJC Scopus subject areas

  • Electrical and Electronic Engineering

Cite this

Naor, A., & Verstraete, J. (2005). Improved bounds on the size of sparse parity check matrices. In 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.

Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05. Vol. 2005 2005. p. 1749-1752 1523645.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Naor, A & Verstraete, J 2005, Improved bounds on the size of sparse parity check matrices. in 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
Naor A, Verstraete J. Improved bounds on the size of sparse parity check matrices. In Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05. Vol. 2005. 2005. p. 1749-1752. 1523645 https://doi.org/10.1109/ISIT.2005.1523645
Naor, Assaf ; Verstraete, Jacques. / Improved bounds on the size of sparse parity check matrices. Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05. Vol. 2005 2005. pp. 1749-1752
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