  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
