On the rank of a random binary matrix

TL;DR

This paper analyzes the rank of random binary matrices with fixed number of ones per column over GF_2, providing asymptotic estimates for rank and minimum basis weight, extending known results for special cases.

Contribution

It offers the first asymptotic estimates for the rank and minimum basis weight of such matrices, generalizing previous specific case results.

Findings

Asymptotic estimate for the rank in terms of c, n, k

Asymptotic estimate for the minimum weight of a basis

Extension of the known result for k=2 to general k

Abstract

We study the rank of the random $n\times m$ 0/1 matrix ${\bf A}_{n,m;k}$ where each column is chosen independently from the set $\Omega_{n,k}$ of 0/1 vectors with exactly $k$ 1's. Here 0/1 are the elements of the field $GF_2$. We obtain an asymptotically correct estimate for the rank in terms of $c,n,k$, assuming that $m=cn$. In addition, we assign i.i.d. $U[0,1]$ weights $X_{{\bf c}},{\bf c}\in\Omega_{n,k}$ and let the weight of a set of columns $C$ be $X(C)=\sum_{{\bf c}\in C}X_{{\bf c}}$. Let a basis be a set of $n-1_{k\text{even}}$ linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight of a basis. This generalises the well-known result for $k=2$ viz. that the expected length of a minimum weight spanning tree tends to $\zeta(3)$.