Bit propagation over a noisy grid

Below is an open problem, an approachable problem, one perhaps you will be the one to solve! We send a bit over a noisy grid, starting from the origin and propagating out as a wave. Can we can recover the original bit when looking at just the wavefront? This is easy to solve for 1D, mostly solved in 2D, and still open in 3D and above. Here is the lecture that introduced the problem to me, as...