On sequences of Toeplitz matrices over finite fields

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