Antenna communication
We consider communicating across the town between buildings (1,000 meters) and across the country between high hills?? (1,000,000 meters).

The bigger the antenna, the better. Lets assume Citizens Band, near 30MHz with wavelength of 10 meters,  and quarterwave verticals of approximately 100 inches.
This size antenna (which needs a "ground counterpoise" of some wires underneath) can be
strung up the wall of your living room; should you lack a cathedral ceiling, simply add some CLCL variable-capacitors fixed-inductors to have a fine resonance at your carrier; this is called antenna-tuner or matching.

Assume 10 bits per second; this allows very low power or very long range. And assume simple ON/OFF modulation, or BPSK, or perhaps differential QPSK which is self-synchronizing (there is no ambiguity). You could go for dense constellations, but the symbol rate becomes very low and phasenoise and carrier acquisition become big problems.

Assume thermal floor at room temperature is the famous -174 dBm/rootHertz which comes from K*T (Boltzmann constant and Kelvin temperature) being exactly 4.00e-21 watts/Hertz at 290 Kelvin.

Assume SignalNoiseRatio of 10dB. Depending on single or double-sided  bandwidth, you will have approximately 0.1% Bit Error Rate BER), or 10^-3 rate. 10dB allows plenty of margin to bring Bit Error Detection and Correction into your system; this reduces lost-packets and the need to re-transmit packets.

Assume (front end losses plus Noise Figure) to total 4dB. This is 1/2.5 loss.

We make no allowance for multipathing, foliage, or interference, or timing-recovery. Nor for long-distance ionospheric reflections.

This is the signal strength we need:

-174    thermal floor at 1 cycle-per-second (1 Hertz) bandwidth, with


10dB  for 10Hz bandwidth
10dB   SignalNoiseRatio (this ignores the wonders of TurboCodes, etc)
4dB   losses and Noise Figure of that first transistor


and all this becomes

-174 + 24 = -150dBm. Given -120dBm across 50 ohms is 0.632 microVolts Peak Peak, we are 30 dB (31.6X) weaker, at 20 nanoVolts Peak Peak. Just a number.

Remember the -150dBm energy floor.

Assume the antennas have gain=1 (compared to an ideal isotropic antenna). So we don't worry about antennas any more.

Now, since the energy spreads out in X and in Y as the RF energy propagates, we have range*range reduction in available energy at the receiver. There also is a hemispheric bit of math, involving 4*PI. 

The result is the PATHLOSS: 22dB + 10*log10[  (range/wavelength)^2 ]

Lets consider two situations:

A) across town 1,000 meters

B) across the country 1,000,000 meters

Across_town: range/wavelength = 1,000/10 = 100

with PATHLOSS (the spreading out of energy in 3_D) of 22+40 == 62dB. Which is a little more than 1Million:1 energy spreading.

Given we need -150 from the across-town receiver antenna, we need -150+62 or -88dBm into the Transmitter antenna in that highrise across the park. How big is this? since 0dBm/50_Ohm is 0.632vPP, and -88dB is near 1/31,000 we need 0.632/31,000 = 20 microVolts PeakPeak into the transmitter antenna.

For a robust link, you should consider pseudo-random spreading at the transmitter, and then a synchronized de-spread and low-pass-filter at 10Hz, in your receiver frontend. These systems may have regulations applying. I don't know.

Now for ACROSS THE COUNTRY. give the across-town math for 1,000 meters, we easily extend this to 1,000,000 meters.

The PATHLOSS increases by 1,000 ^2 or one Million X power, or 1 Thousand X voltage. Thus you need 20 milliVolts Peak Peak, at Citizens Band.

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So what are we learning here? With physically large antennas for long wavelengths and low carrier frequencies, and using low low symbol/bit/baud rates, you can use very low power.

However, you must design the system to be able to track (search for carrier) transmitter frequency wander, caused by cheap XTAL oscillators. 

You must tolerate enormous blockers either on top of your signal, or at some small offset frequency. In either case,  you need receiver front-end amplifiers with high overload specs, such as the IP2 and IP3 etc. This requires high currents in the front-end LNAs (or transformer feedback to linearize). Also to handle blockers on top of your signal (either shared-channel or otherwise), you should examine code-spreading at TX, and then correlations to de-spread in the Receiver. A code-spread ratio of 1,024:1 should give you a 60dB better tolerance for energy on top of your channel.
You must still have a very high purity (very low distortion) front end in the receiver. By the way, vacuum tubes may inherently provide the needed performance (except for the thermal noise floor?). With transistors, expect to operate at high currents and use discrete resistors to linearize the operating point.

PS there may be errors herein.