Periodic coset state
= Periodic coset state
For $r\mid N$, a periodic coset state in the computational basis of $\mathbb Z_N$ is
$$
\frac1{\sqrt{N/r}}\sum_{j=0}^{N/r-1}|x_0+jr\rangle.
$$
It is the <coset state> of the subgroup $r\mathbb Z_N$ translated by $x_0$.