Consider a rectangular grid graph of size $m \times n$, like following $4 \times 7$:
Fix $m \geq 2$, I'm trying to proof that exist a positive constant $\epsilon > 0$ such that the number of cycles of the rectangular grid of size $m \times n$ with $n \geq 2$ named $C(m,n)$ is at least $2^{\epsilon \cdot n}$, i.e. $C(m,n) \in 2^{\Omega(n)}$.
Idea: My conjecture is that $\epsilon =1$ and to demonstrate this I'm thinking in assign to each binary number of lenght $n$ a specific cycle in the grid such that the correspondence is injective and then I will have $C(m,n) \geq 2^n$ for every $n$ but I'm stack. Any hint?
Edit: $m$ and $n$ can be equal (square grid) and as big if it is necessary.
Here's a proof without words for $2^n$ cycles