It is well known that the projective plane of order $2$ can be represented by the circulant matrix $M_2:=circ(x,x,1,x,1,1,1)= \begin{pmatrix} x&x&1&x&1&1&1\\ 1&x&x&1&x&1&1\\ 1&1&x&x&1&x&1\\ 1&1&1&x&x&1&x\\ x&1&1&1&x&x&1\\ 1&x&1&1&1&x&x\\ x&1&x&1&1&1&x\\ \end{pmatrix}.$

For the one of order $3$ we can take $M_3:=circ(x,x,1,1,x,1, x, 1, 1, 1,1,1,1)= \begin{pmatrix} x& x& 1& 1& x& 1& x& 1& 1& 1& 1& 1& 1\\ 1& x& x& 1& 1& x& 1& x& 1& 1& 1& 1& 1\\ 1& 1& x& x& 1& 1& x& 1& x& 1& 1& 1& 1\\ 1& 1& 1& x& x& 1& 1& x& 1& x& 1& 1& 1\\ 1& 1& 1& 1& x& x& 1& 1& x& 1& x& 1& 1\\ 1& 1& 1& 1& 1& x& x& 1& 1& x& 1& x& 1\\ 1& 1& 1& 1& 1& 1& x& x& 1& 1& x& 1& x\\ x& 1& 1& 1& 1& 1& 1& x& x& 1& 1& x& 1\\ 1& x& 1& 1& 1& 1& 1& 1& x& x& 1& 1& x\\ x& 1& x& 1& 1& 1& 1& 1& 1& x& x& 1& 1\\ 1& x& 1& x& 1& 1& 1& 1& 1& 1& x& x& 1\\ 1& 1& x& 1& x& 1& 1& 1& 1& 1& 1& x& x\\ x& 1& 1& x& 1& x& 1& 1& 1& 1& 1& 1& x \end{pmatrix}.$

- Can the incidence structure of a finite projective plane always be written as a circulant matrix?
- Is this known at least for the Desarguesian planes?
- Is this circulant matrix essentially unique for a given plane? (i.e. up to cyclic permutation and reflection)

The determinant of this matrix for any projective plane of order $d$ is $$\det M_d=d^{d(d+1)/2}(x-1)^{d(d+1)}[(d+1)x+d^2],$$ but the structure of this expression does not seem to give more hints.