In the Wikipedia article on circulant matrices, circulant matrices are given as follows:
\begin{equation} C= \begin{bmatrix} c_0 & c_{n-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{n-1} & & c_{2} \\ \vdots & c_{1}& c_0 & \ddots & \vdots \\ c_{n-2} & & \ddots & \ddots & c_{n-1} \\ c_{n-1} & c_{n-2} & \dots & c_{1} & c_0 \\ \end{bmatrix} \end{equation}
Now, in the following finite state machine, an interesting matrix comes up. Imagine we have a binary array of fixed width, say, 3, and we number the possible states of that array from 0-7. Each time we want to advance state, we drop the left-most digit from the array and push either a zero or one onto the right, forming a new 3-digit binary number. So each state has only two possible destination states based on this evolution.
The adjacency matrix for such a state machine looks like this (zeros omitted):
\begin{equation} A = \begin{bmatrix} 1 & 1 &&&&&& \\ && 1 & 1 &&&& \\ &&&& 1 & 1 && \\ &&&&&& 1 & 1 \\ 1 & 1 &&&&&& \\ && 1 & 1 &&&& \\ &&&& 1 & 1 && \\ &&&&&& 1 & 1 \end{bmatrix} \end{equation}
Do people call this a "block circulant matrix" or something? Is there any literature? I think the "width" of the rotated block corresponds to the arity of the field (so for ternary digits the width would be 3 instead of 2), and the matrix dimensions would be $3^3 = 27$ instead of $2^3 = 8$.