[jdietrich]

The Shannon-Hartley theorem is your friend. The theoretical maximum data rate of a channel is the product of bandwidth and SNR. With a sufficiently low data rate and/or a sufficiently wide bandwidth, we can effectively communicate at well below the noise floor.

There are many propagation modes to choose from. HF frequencies (~3MHz to ~30MHz) will propagate via the skywave mode, bouncing off charged layers in the ionosphere. By using codings with extremely low data rates (JT65, QRSS), amateurs routinely make contacts over thousands of miles on a fraction of a watt. Unfortunately, this mode relies on sunspot activity and we're currently at a minimum in the solar cycle. Frequencies below 3MHz will diffract around obstacles and follow the curvature of earth; unfortunately an efficient antenna at these wavelengths is enormously long, so a small system would have absolutely vast antenna losses. NVIS propatation is usable between about 3 and 8MHz, although you're limited to about 600km in a single hop. Multi-hop propagation is possible, although the path loss increases exponentially. Satellite is the other obvious propagation mode and is surprisingly accessible to amateurs.

Solutions to the Shannon-Hartley equation that involve very wide bandwidths are sadly underexplored by the amateur community, because of an FCC requirement to use the narrowest possible bandwidth and the relatively meagre frequency allocations available to amateur operators. The extraordinarily wide bandwidth of a modern direct-conversion transceiver offers some tantalising possibilities.