On sequences of Toeplitz matrices over finite fields

TL;DR

This paper investigates the nullity patterns of sequences of Toeplitz matrices over finite fields, providing counting formulas and an elementary proof related to the number of such matrices with specified rank.

Contribution

It introduces a detailed analysis of nullity sequences of Toeplitz matrices over finite fields and offers a new elementary proof for counting matrices of a given rank.

Findings

Derived formulas for counting Toeplitz matrices with specific nullity patterns

Established a connection between nullity patterns and matrix rank over finite fields

Provided an elementary proof for the enumeration of Toeplitz matrices of any given rank

Abstract

For each non-negative integer $n$ let $\mathcal{A}_n$ be an $n+1$ by $n+1$ Toeplitz matrix over a finite field, $F$, and suppose for each $n$ that $\mathcal{A}_n$ is embedded in the upper left corner of $\mathcal{A}_{n+1}$. We study the structure of the sequence $\nu = \{\nu_n :n \in \mathbb{Z}^+\}$, where $\nu_n = \text{null}(\mathcal{A}_n)$ is the nullity of $\mathcal{A}_{n}$. For each $n\in \mathbb{Z}^+$ and each nullity pattern $\nu_0,\nu_1,\dots,\nu_n$, we count the number of strings of Toeplitz matrices $\mathcal{A}_0,\mathcal{A}_1,\dots,\mathcal{A}_{n}$ with this pattern. As an application we present an elementary proof of a result of D. E. Daykin on the number of $n\times n$ Toeplitz matrices over $GF(2)$ of any specified rank. (This is a corrected version of the paper published in Linear Algebra and Its Applications $561$ \, $(2019), 63-80$.) 2000 MSC Classification 15A33, 15A57