Circulant matrices are very useful in digital image processing. I found the general formula for determinant of circulant matrix. But I think it is not suitable for block-circulant matrices.

For example, consider the formula for $\det(K)$,

where $$K = \left(\begin{array}{cccc} A & B & C & D \\ D & A & B & C \\ C & D & A & B \\ B & C & D & A \end{array}\right) $$

and $A$ , $B$ , $C$ and $D$ are size $n \times n$.