The rank of sparse random matrices

We determine the asymptotic normalized 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 (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. MSC: 05C80, 60B20, 94B05 Coja-Oghlan supported by DFG CO 646/3 and 646/4.