[Paper Review] Weight and rank of matrices over finite fields

Theresa Migler, Kent E. Morrison+1 more|ArXiv.org|Mar 18, 2004

graph theory and CDMA systems22 citations

TL;DR

This paper derives a closed-form expression for the average weight of $m \times n$ matrices of rank $k$ over a finite field $\mathbb{F}_q$, showing it equals $mn(1 - 1/q)$. For rank-one matrices, it fully characterizes the weight distribution and proves a central limit theorem, demonstrating convergence to a normal distribution as $m$ or $n$ grows large.

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.

Motivation & Objective

• To determine the average weight of $m \times n$ matrices of rank $k$ over $\mathbb{F}_q$, a fundamental statistical property in random matrix theory over finite fields.
• To characterize the complete weight distribution for rank-one matrices over finite fields, which remains an open problem for higher ranks.
• To establish a central limit theorem for the weight of rank-one matrices, showing asymptotic normality as matrix dimensions grow.
• To provide a foundation for understanding the joint distribution of rank and weight in random matrices over finite fields, a key challenge in coding theory and combinatorics.

Proposed method

• Uses combinatorial enumeration of matrices via $q$-binomial coefficients to count matrices of given rank.
• Applies reduced row echelon form (RREF) to factorize any rank-$k$ matrix as $A = CR$, where $C$ is $m \times k$ of full rank and $R$ is $k \times n$ in RREF.
• Leverages symmetry and row/column permutations to show that the probability $\mathbf{P}(a_{ij} \neq 0)$ is uniform across all entries for fixed rank $k$, enabling average weight computation.
• For rank-one matrices, models the weight as the product of two independent binomial random variables: one for the non-zero rows of $C$ and one for the non-zero entries of $R$, both conditioned on non-vanishing.
• Applies the Central Limit Theorem to the product of these binomial variables, proving asymptotic normality of the weight distribution as $m$ or $n$ tends to infinity.
• Derives exact expressions for the mean and variance of the weight of rank-one matrices, showing the variance scales as $O(n^2)$ when $m \approx n$.

Experimental results

Research questions

• RQ1 What is the average weight of an $m \times n$ matrix of rank $k$ over $\mathbb{F}_q$?
• RQ2 What is the complete distribution of the weight for rank-one matrices over finite fields?
• RQ3 Does the weight distribution of rank-one matrices converge to a normal distribution as the matrix dimensions grow?
• RQ4 How does the joint distribution of rank and weight behave in random matrices over finite fields?
• RQ5 Can the probability that a given entry is non-zero be uniformly bounded across all positions for fixed rank matrices?

Key findings

• The average weight of an $m \times n$ matrix of rank $k$ over $\mathbb{F}_q$ is $mn(1 - 1/q)$, independent of $k$, due to uniform entry-wise non-zero probability.
• For rank-one matrices, the weight distribution is the product of two independent binomial variables conditioned to be positive, with probability mass function $\mathbf{P}(\mathrm{wt}\,A = \omega) = \sum_{\mu\nu=\omega} \binom{m}{\mu} \binom{n}{\nu} \frac{(q-1)^{m+n-\mu-\nu} q^{\mu+\nu}}{(q^m - 1)(q^n - 1)}$.
• The mean weight of a rank-one matrix is $mn(1 - 1/q)^2$, and the variance is asymptotically $mn(1 - 1/q)^2 + mn(m+n-2)(1 - 1/q)^3 + m(m-1)n(n-1)(1 - 1/q)^4 - (mn(1 - 1/q))^2$.
• As $m$ or $n$ tends to infinity, the weight distribution of rank-one matrices converges in distribution to a normal law, with mean $\approx mn(1 - 1/q)^2$ and standard deviation $\sim O(n)$ when $m \approx n$.
• The probability that a specific entry is non-zero in a rank-$k$ matrix is $1 - 1/q$, and this is uniform across all positions due to symmetry under row and column permutations.
• The weight of a rank-one matrix cannot exceed $mn$, but not all weights in $[1, mn]$ are possible, as the weight is the product of row and column non-zero counts, leading to gaps and spikes in the distribution.