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 conje
cture 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.
A graph $G$ whose edges are coloured (not necessarily properly) contains a full rainbow matching if there is a matching $M$ that contains exactly one edge of each colour. We refute several conjectures on matchings in hypergraphs and full rainbow matc
hings in graphs, made by Aharoni and Berger and others.
