# The rank of sparse random matrices

## TL;DR

This paper determines the rank of sparse random matrices over any field with fixed row and column sparsity, providing a formula for low-density parity check codes and confirming Lelarge's conjecture.

## Contribution

It introduces a new method involving random perturbation to analyze the rank of sparse matrices, applicable across various fields.

## Key findings

- Derived a formula for the rank of sparse random matrices
- Confirmed Lelarge's conjecture on LDPC code rates
- Developed a novel perturbation technique for matrix analysis

## 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.