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.

翻译：我们确定了任意域上随机矩阵的秩，该矩阵的每行和每列具有指定数量的非零条目。作为应用，我们得到了低密度奇偶校验码的速率公式。该公式证实了Lelarge（2013）的一个猜想。证明基于耦合论证和一种适用于任何矩阵的新型随机扰动方法，该方法能减少短线性关系的数量。