Take a function $f: Z_N\rightarrow R$. Construct an $N \times N$ matrix where the $(i,j)$th element of the matrix is $f(i-j)$, where $i-j$ is interpreted mod $Z_N$. The resulting matrices are precisely the circulant matrices.
Suppose, instead, that $N=2^n$. We could perform the analogous construction where we interpret $i-j$ over $Z_2^n$. Or, to paraphrase, we interpret $i$ and $j$ as strings of $n$ bits and put $f(i\oplus j)$ in the $(i,j)$th entry of the matrix, where $\oplus$ is a bitwise XOR.
My question is: is there a standard name for the second class of matrices?
Or, in pictures, if this is a circulant matrix: $$\begin{bmatrix} a&b&c&d&e&f&g&h\\ h&a&b&c&d&e&f&g\\ g&h&a&b&c&d&e&f\\ f&g&h&a&b&c&d&e\\ e&f&g&h&a&b&c&d\\ d&e&f&g&h&a&b&c\\ c&d&e&f&g&h&a&b\\ b&c&d&e&f&g&h&a \end{bmatrix}$$
Then what is this matrix? $$\begin{bmatrix} a&b&c&d&e&f&g&h\\ b&a&d&c&f&e&h&g\\ c&d&a&b&g&h&e&f\\ d&c&b&a&h&g&f&e\\ e&f&g&h&a&b&c&d\\ f&e&h&g&b&a&d&c\\ g&h&e&f&c&d&a&b\\ h&g&f&e&d&c&b&a\\ \end{bmatrix} $$ (The motivation arose from looking for a simple class of matrices that share the same eigenvectors, like circulant matrices do. The matrices above seem to relate to the Hadamard transform in the same way that circulants do to the discrete Fourier transform.)
