A note on the rank of a sparse random matrix

Abstract

Let $\mathbf{A}_{n,m;k}$ be a random $n \times m$ matrix with entries from some field $\mathbb{F}$ where there are exactly $k$ non-zero entries in each column, whose locations are chosen independently and uniformly at random from the set of all ${n \choose k}$ possibilities. In a previous paper (arXiv:1806.04988), we considered the rank of a random matrix in this model when the field is $\mathbb{F}=GF(2)$. In this note, we point out that with minimal modifications, the arguments from that paper actually allow analogous results when the field $\mathbb{F}$ is arbitrary. In particular, for any field $\mathbb{F}$ and any fixed $k\geq 3$, we determine an asymptotically correct estimate for the rank of $\mathbf{A}_{n,m;k}$ in terms of $c,n,k$ where $m=cn/k$, and $c$ is a constant. This formula works even when the values of the nonzero elements are adversarially chosen. When $\mathbb{F}$ is a finite field, we also determine the threshold for having full row rank, when the values of the nonzero elements are randomly chosen.