M911

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). Moreover we derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and {0, 1}-matrices over the rationals.

Maurice Rolvien

TU Dortmund