The rank of sparse random matrices
Combinatorics
2024-06-21 v4 Discrete Mathematics
Probability
Abstract
We determine the rank of a random matrix 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 (2013). The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations.
Cite
@article{arxiv.1906.05757,
title = {The rank of sparse random matrices},
author = {Amin Coja-Oghlan and Alperen A. Ergür and Pu Gao and Samuel Hetterich and Maurice Rolvien},
journal= {arXiv preprint arXiv:1906.05757},
year = {2024}
}
Comments
This article supersedes arXiv:1810.07390