![Geometry and Computation in High Dimensions.png](/sites/default/files/styles/workshop_banner_sm_1x/public/2023-05/Geometry%20and%20Computation%20in%20High%20Dimensions.png.jpg?itok=1JtiYLWR)
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.