Adding Vectors

Geodude wants to assign one of the integers $1,2,3,\ldots,11$ to each lattice point $(x,y,z)$ in a 3D Cartesian coordinate system. In how many ways can Geodude do this if for every lattice parallelogram $ABCD$, the positive difference between the sum of the numbers assigned to $A$ and $C$ and the sum of the numbers assigned to $B$ and $D$ must be a multiple of $11$? (A lattice point is a point with all integer coordinates. A lattice parallelogram is a parallelogram with all four vertices lying on lattice points. Here, we say four not necessarily distinct points $A,B,C,D$ form a parallelogram $ABCD$ if and only if the midpoint of segment $AC$ coincides with the midpoint of segment $BD$.)

Source: 2012-2013 Winter Online Math Open