Weight and rank of matrices over finite fields

TL;DR

This paper derives a formula for the average weight of matrices with a fixed rank over finite fields and provides a detailed analysis for rank-one matrices, including their weight distribution and a central limit theorem.

Contribution

It introduces a closed-form expression for the average weight of matrices of a given rank over finite fields and characterizes the weight distribution for rank-one matrices.

Findings

Closed-form formula for average weight of rank-k matrices

Complete weight distribution for rank-one matrices

Central limit theorem for rank-one matrix weights

Abstract

Define the weight of a matrix to be the number of non-zero entries. One would like to count $m$ by $n$ matrices over a finite field by their weight and rank. This is equivalent to determining the probability distribution of the weight while conditioning on the rank. The complete answer to this question is far from finished. As a step in that direction this paper finds a closed form for the average weight of an $m$ by $n$ matrix of rank $k$ over the finite field with $q$ elements. The formula is a simple algebraic expression in $m$, $n$, $k$, and $q$. For rank one matrices a complete description of the weight distribution is given and a central limit theorem is proved.