$N(N+1)$ lines, that is, solutions of linear equations in $\alpha_1$ and $\alpha_2$, and these lines can be sorted into $N+1$ sets, each set consisting of $N$ parallel lines. We call a complete set of parallel lines a “striation.”

For any density matrix $\hat{\rho}$, the corresponding Wigner function is defined as $$\label{W}
W_\alpha = \frac{1}{N}\, \hbox{Tr} (\hat{A}_\alpha \hat{\rho}),$$ where the operators $\hat{A}_\alpha$ are given as follows. For $N=2$, $$\label{A2}
\hat{A}_\alpha = \frac{1}{2}[\hat{I} + (-1)^{\alpha_1} \hat{\sigma}_z + (-1)^{\alpha_2} \hat{\sigma}_x + (-1)^{\alpha_1 + \alpha_2} \hat{\sigma}_y],$$ where $\hat{I}$ is the $2 \times 2$ identity matrix and the $\hat{\sigma}$’s are the Pauli matrices. For prime $N$ greater than 2, we write $\hat{A}_\alpha$ in terms of the phase space displacement operators $\hat{D}_\beta$, defined by $$\label{displacements}
\hat{D}_\beta = \omega^{\beta_1 \beta_2/2} \hat{X}^{\beta_1} \hat{Z}^{\beta_2}.$$ Here $\omega = e^{2 \pi i/N}$ and the arithmetic in its exponent is understood to be mod $N$. (So $\omega^{1/2} = \omega^{(N+1)/2}$.) The basic displacement operators $\hat{X}$ and $\hat{Z}$ are defined in terms of a standard orthonormal basis $\{|q\rangle \}$ as [@Weyl1950] $$\begin{split}
&\hat{X} |q\rangle = |q+1 \; \hbox{(mod $N$)}\rangle \\
&\hat{Z} |q\rangle = \omega^q |q\rangle.
\end{split}$$ Now we define the operators $\hat{A}_\alpha$ by $$\label{DAs}
\hat{A}_\alpha = \frac{1}{N} \sum_\beta \hat{D}_\beta \omega^{\langle \alpha, \beta \rangle},$$ where 