Proceedings
Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

The rank of sparse random matrices

Abstract

We determine the rank of a random matrix A over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula vindicates a conjecture of Lelarge [Proc. IEEE Information Theory Workshop 2013]. The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that likely diminishes the number of short linear relations.

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cover image Proceedings
Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 579 - 591
Editor: Shuchi Chawla
ISBN (Online): 978-1-611975-99-4

History

Published online: 23 December 2019

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*
Coja-Oghlan's research is supported by DFG CO 646/4. Ergür's research is supported by Einstein Foundation, Berlin. Gao's research is supported by ARC DE170100716 and ARC DP160100835.

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