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If you had two computers with two atomic clocks you could measure the time between signals at $10^{-16}\,\text{s}$. Can the time between signal pulses be used to encode additional bits of information over and above that encoded in the standard way?

If a channel is transmitting bits at $10^6\,\frac1{\text{s}}$ one might imagine that the average time between signals is one cycle length - given random data. So for each $10^{-6}\,\text{s}$ you will also be able to encode and additional $10^{16-6} = 10^{10} \approx 2^{34}$ – i.e. an additional 34 bits for every ordinary bit.

Does this come with obvious downsides? Is it already used?

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From the DSP point of view it is possible, and if you produce and detect pulses this fast you could also transmit $10^{16}$, what is the minimum/maximum interval between two pulse?

Let's assume you have a distribution for the pulse length $p_k = p(k\, T)$. The maximum information can be transmitted per pulse is given by the entropy of this distribution.

$$ -\sum p_k \log_2(p_k) $$

the pulse transmision rate is the inverse of the expected duration of a pulse

$$ \frac{1}{\sum k \, p_k} $$

The product of these two quantities give you the maximum information you can transmit.

$$ \frac{-\sum p_k \, \log_2(p_k)}{\sum k \, p_k} $$

Then you must optimize this subject to the restrictions of the physical system, if you can transmit have any probability for any $k \ge 1$, the best $p_k$ distribution the $1/2^k$ and will give you 1-bit per $T$ in your case $10^{16}$ bit per second.

Maybe you want to include some noise to the pulse position measurements, then you have to compute the mutual information, that will be slightly less than the source Entropy.

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There is no free lunch here.

Can the time between signal pulses be used to encode additional bits of information over and above that encoded in the standard way?

I believe, this assumes that there are only very short signal pulses and the signal is zero otherwise. That's typically not the case in a "normal" communication system. The communication signal just transitions between the current and the next symbol value and the speed of the transition is limited by the bandwidth of the channel.

In order to use the symbol timing to transmit more information, the receiver would needs a way to recognize when exactly the symbols is happening and the transmitter needs to be able to transition between symbols as fast as the timing modulation depth requires it.

Both are solvable problems but the resulting requirements in terms of bandwidth and noise floor makes in most cases for a less effective communication system than a "standard" solution.

Roughly speaking: if your channel has enough bandwidth that you can modulate the symbol timing by by +-25%, you are better off using the extra bandwidth just to double the symbol rate.

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