Let $C(x,y)$ be a circle passing through the points $(x,y)$, $(x,y+1)$, $(x+1,y)$ and $(x+1,y+1)$.

For positive integers $m$ and $n$, let $E(m,n)$ be a configuration which consists of the $m\cdot n$ circles:
$\{C(x,y):0\le x<m,0\le y<n,x\text{ and }y\text{ are integers}\}$.

An Eulerian cycle on $E(m,n)$ is a closed path that passes through each arc exactly once.
Many such paths are possible on $E(m,n)$, but we are only interested in those which are not self-crossing: a non-crossing path just touches itself at lattice points, but it never crosses itself.

The image below shows $E(3,3)$ and an example of an Eulerian non-crossing path.

Let $L(m,n)$ be the number of Eulerian non-crossing paths on $E(m,n)$.
For example, $L(1,2)=2$, $L(2,2)=37$ and $L(3,3)=104290$.

Find $L(6,10)mod{10}^{10}$.