Coja-Oghlan, AminErgür, Alperen A.Gao, PuHetterich, SamuelRolvien, Maurice2024-02-212024-02-212022-04-23http://hdl.handle.net/2003/42342http://dx.doi.org/10.17877/DE290R-24179We determine the asymptotic normalized rank of a random matrix A over an arbitrary field with prescribed numbers of nonzero 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.enFinite fieldRandom constraint satisfactionRandom matricesRankSparse matrices004Text