How many photons are received per bit transmitted from Voyager 1?

Didn’t realize the math would be that straightforward. Is there something the author isn’t taking into account or is that a decent plausible range?

Seems like we can just build a bigger receiving dish

Not my field, but assuming transmitting hardware (including beam forming) is constant and that atmosphere can mostly be ignored (see comments about it usually being a non-impact in the transmission frequencies), two approaches would suggest:

1. Increase the effective receiving dish size, to capture more of the signal. Essentially, this would be effective in direct proportion to beam spread (the more beam spread, the bigger dish you can use to capture signal).

In practice, this would use multiple geographically-displaced dishes to construct a virtually-larger dish, to allow for better noise-cancellation magic (and at lower cost than one huge dish). I believe the deep space network (DSN) already does this? Edit: It certainly has arrayed antennae [0], though not sure how many are Voyager-tasked.

2. Increase the resilience of the signal, via encoding. The math is talking about bits and photons, but not encoded information. By trading lower bit-efficiency for increased error tolerance (i.e. including redundant information) we can extract a coherent signal even accounting for losses.

Someone please point out if I'm wrong, but afaik the Shannon–Hartley limit speaks to "lower" in the physical stack than error coding. I.e. one can layer arbitrary error coding on top of it to push limits (at the expense of rate)?

If the above understanding is correct, is there a way to calculate maximum signal distance assuming a theoretically maximally efficient error coding (is that a thing?) ? Or is that distance effectively infinite, assuming you're willing to accept an increasingly slow bit receiving rate?

[0] https://en.m.wikipedia.org/wiki/NASA_Deep_Space_Network#Ante...

Maybe multiple receiving antennas could be used, spaced widely apart (e.g. one on Earth, one on the moon, one in orbit somewhere); each would receive some signal affected by noise, but the noise would be different for each (such as the atmosphere affecting only the on-ground antenna). Also given that we know the precise location of the transmitter, we can work-out the exact phase difference between the antennas. Some digital post-processing would combine the raw signal+noise from the multiple receivers and extract the signal.

In fact, I assume that at this distance, even a very narrow signal would spread wide enough to illuminate more than just the Earth diameter.