Like https://en.wikipedia.org/wiki/Lights_Out_%28game%29 ?

But I don't see how your explanation is compatible with their claim that found a neat structure like a "doubly even self-dual linear" ECC. If their claim is that all transitions maintain that structure, then you are guaranteed to stay inside the structure no matter how long you wander around the graph.

I was thinking more like a Drunkard's Walk[1]. Eventually, every possible location (state) will be visited.

[1] http://en.wikipedia.org/wiki/Random_walk

If you have an integer random walk and the step sizes are multiples of 3, you cannot reach a location that is not a multiple of 3. The ECC structure they described is similar to the set of multiples of 3: for example, an analog computer trying to simulate such a walk would periodically round the location to the nearest multiple of 3 to counter the effects of noise. The rounding does not change correct computations (becase they would already be multiples of 3), but it increases the likelihood that the result of the computation is correct.

Right, but here, each half of the graph is a different code. If you screw up a "white dot", you just get a "black dot".