The exposition is that a civilization exists about 200 light years from earth and has been capable of radio communication for several thousands of years. They are not trying to hide from detection but have not been actively trying to communicate either.
I know we have several programs going on to try and detect such extraterrestrial radio signal. What plausible astronomical, technological, or human-error-related factor could I use to explain humanity not having picked up their signal?
There are about 134,000 stars within 200 light years of Earth. That's a lot of stars. We can't look at even a significant fraction of all of them closely enough to detect the equivalent of human-made radio waves.
A 200 ly radius is a sphere 33,493,333 ly3 in volume. The stellar density near Earth is about 0.004/ly3. Put them together and you get about 134,000 stars.
This answer on Physics.SE says we're just getting to the point where we could see radio and TV transmissions from the Earth at 200 light years away. Unless the aliens happen to point a directional transmitter right at the Earth, we're not going to pick up their omni-directional broadcasts that happen to go off into space. These diffuse as the square of the distance, so their detectability drops of very rapidly.
Spotting a radio signal 200 ly out requires a very expensive, high gain radio antenna. That sort of sensitivity requires it to be highly directional, so you can't look at the whole sky at once. SETI choses particularly likely stars and likely bands. Project Phoenix did just this in the late 90s. They picked 800 "likely" stars within a 200 ly range or about 0.6% of all the stars in that range.
Your civilization's star might not match SETI's idea of a star that's worth looking at. Or they might be transmitting in frequency ranges ranges that SETI doesn't think to look at. Or it might have simply been bad luck that their star didn't get on SETI's list. Or, as others have correctly suggested, they're not inefficiently broadcasting omnidirectional high power radio waves anymore.
People tend to underestimate just how hard it is to detect radio transmissions from a distant source.
When a spacecraft communicates with the Earth using dish antennas on both ends, the received signal strength for a pure sinewave (unmodulated) carrier can be calculated as
$$P_R=\frac{P_TA_TA_R}{\lambda^2R^2},$$
where $P_T$ is the transmitter power, $A_T$ is the transmitter antenna's effective surface area, $A_R$ is the same for the receiver, $\lambda$ is the wavelength and $R$ is the distance between the transmitter and receiver.
By way of example, if I plug in the numbers for Pioneer 10 near the end of its mission (8 W transmitter, 80 AU from the Earth, a transmission frequency of roughly 2.3 GHz, a transmitter antenna diameter of 2.75 m, and 70 m dish antenna on the ground), I get a value of -165 dBm (decibel over milliwatts).
So now let me move the signal source from 80 AU to, say, 120 light years, which is a 100,000-fold increase. This reduces the signal strength by a whopping 100 dB. Further, let me assume that it is not transmitted by a dish transmitter aimed directly at the Earth (why would it be?) but rather, it is omnidirectional. At the wavelengths used by Pioneer, that's another ~30 dB decrease in the signal strength. On the other hand, let me assume that the transmitter is a lot more powerful, say, 8 MW instead of 8 W. That is a 60 dB increase in signal strength.
So a 70 m dish antenna (the largest used by NASA's Deep Space Network) would now see a signal at -245 dBm. Or rather, it would not see that signal; it is orders of magnitude below its detection threshold (maybe around -195 dBm).
OK, forget the DSN. Let's say we use the new Chinese radio telescope, FAST. Its dish is a whopping 500 meters in diameter. That is a 17 dB boost in the received signal level. Let me be generous and bump it to 20 dB; so now we are getting a -225 dBm signal.
It is still some three orders of magnitude less than what we can actually detect. And remember, this is just to detect the presence of a signal. Any modulation on that signal reduces the signal level, and detecting the modulation itself, even at a very low data rate, would be still orders of magnitude away.
So no, contrary to popular belief, civilizations that use radio are not shining beacons in the (radio) sky. Aliens won't be beaming back to us Hitler's 1936 Olympic opening speech from dozens of light years away. These "stray" radio broadcasts are immensely difficult simply to detect (never even mind decode or demodulate). And it's not a limit of technology, these are hard physical limits; an antenna of a given size only intercepts so many photons, so much radio energy, even if otherwise it is a "perfect", flawless instrument.
Quite simply because they've had the technology for thousands of years. Their compression and encoding has progressed past the point where we can identify it as a coherent signal.
There's a tiny window where their broadcast technology would be comprehensible to our receivers during which we could possibly pick up their signal. That would be at the point only recently after they'd invented radio, but before compression and digital encoding. Even if we'd picked up a signal from that point, the chances are we wouldn't recognise it simply due to different technologies.
However that window is out by thousands of years in this case, and it's unlikely that such a technological coincidence would occur in tandem with a grossly overpowered domestic radio broadcast signal.
It's not missing their existence you have to justify, it's detecting them. Our chances of spotting them even under the best of circumstances are minuscule.
Being the more advanced race, they'd have a window for detecting us starting around now and lasting for a few years, but if they don't pick us up at the start of the window, our signals will start to be too complex for them to recognise.
Transmitting radio into space is just a waste of energy, so they shape the lobes of the transmissions carefully, making sure as much as possible is sent to the receivers. This they do by either sending directly at receivers, or for broadcasting they transmit only in the ground plane.
Already today we have WiFi routers that do beam shaping to optimize transmission. And this is after having radio for just about a hundred years.
You don't need anything. Our technology could not pick up our own radio emissions at a 200 light-year distance, so there's no reason to think that we should be able to pick up broadcasts from an alien civilisation. The various SETI programmes are looking for directed transmissions aimed specifically at us, not broadcasts.
Essentially we cannot detect the radio communications of a civilization at 200 light years.
Michael Kjörling's answer to the question How far away would an alien civilization need to be for us to not notice them is very interesting and has plenty of equations. Money quotation:
At current levels of sensitivity, targeted microwave searches could detect the equivalent power of strong TV transmitters at a distance of 1 light year (within which there are no other stars).
The conclusion of the answer is that we won't detect them unless they point a transmitter with enormous power, i.e., comparable to the total electric power produced on Earth, straight at us, and we happen to look for it at the right time on the right frequency.
The common view is that civilian radio transmissions not intended to be receivable over interstellar distances are simply not detectable. On the contrary, high-power directional signals, such as those generated by radiotelescopes (when used as transmitters) or by military radar installations, are detectable over huge distances, but the probability of their detection is very low because they cover a narrow cone and are emitted for a very short time; see "The Benefits and Harm of Transmitting Into Space" by Jacob Haqq-Misra et al. (2013).
The big factor in the detection of alien civilizations is of course the span of time over which they use high-powered radio. See "Calculating the probability of detecting radio signals from alien civilizations" by Marko Horvat; the conclusion of that article is that unless there are very many alien civilization or somehow the aliens develop their technology very much slower than us then the probability of detecting an alien civilization in the galaxy is very small.
"Several thousand" means at least two thousand. Since they are only 200ly away, this means that their first radio signals (which were likely close to what we would consider "radio") are already history. What we receive now, if we receive anything, is 1700+ years ahead of our state of development.
Over two millenia of "radio availability" makes it likely enough that their signals are at the very least something like a forward error-corrected and compressed digital (or trigital, who said an alien bit can't have three states?) signal. Forward error correction would mean they could send a lot more energy-efficiently, which is good for them but not so good for you trying to receive their signals from far away, and compression would mean that whatever you receive, if you receive something, looks like noise.
They might as well use a form of "radio" that we are unfamiliar with (unusual high frequencies?) or entirely unable to interprete. Their form of radio might even be some form of quantum communication, which by design is only receivable by designated subscribers and makes "eavesdropping" from the outside impossible. Quantum communication is something we are presently almost getting to work with photons (well, saying "almost" is wrong... it works, it just isn't ready for mainstream yet). An advanced alien race might as well have some form of "quantum radio" with designated recipients, why not.
They actually tried to communicate (not with us, but maybe with a starship of theirs), and we actually did receive their message. Unluckily, we were unable to decipher or identify it as what it was, nor answer to it. Eventually, the signal was considered "just noise", or something a comet produced.
An entirely different, more conservative, and plausible explanation would be ionospheric (or similar, whatever they have on their planet) reflection, in the same way as shortwave radio is reflected on our planet. This isn't even limited to the planet's atmosphere, their solar system could have an Oort cloud which contains some kind of particles that reflect 99% of all transmissions. Thus, they could even communicate with space ships within their solar system, or a colony on another planet, and we would still have trouble catching a glimpse.
Black matter might be another perfectly plausible reason why we didn't spot them. We know with rather good certitude that such a thing as black matter exists (though we don't really know what it is exactly, or how exactly it complies with our known laws of physics, but we are confident something is there). There might just happen to be some black matter that, for some reason, absorbs radio signals in between us and the aliens. Most radio signals broadcast quasi-light-like, which means if something is in between sender and receiver, the signal is partially or completely obstructed. That is the main reason why TV stations (or GSM stations) have their antennas on high towers or mountains -- their range is primarily limited by obstructing objects and finally Earth's curvature, not so much by the transmitter's power. It is also the reason why orbiting the moon is a no-fun experience for astronauts (there's no such thing as telling "Houston, we have a problem" with the Moon in between).
Finally, even without any special conditions, detecting another civilization which isn't actively trying to communicate is a daunting task, to say the least. Or, you could say, pretty hopeless.
There's the thing about arc length. For something hundreds of light years away, you have to aim darn precisely, and you had better have a really, really directional antenna. But this is not the biggest problem.
The transmitter of a non-trivial nationwide TV station, which is among the strongest "normal" radio emitters that we have (about 2,000 times stronger than e.g. a typical GSM base) typically has EIRPs in the 100kW range. "Kilowatt" sounds like a lot, but it is really a quite small number.
The radiant flux density is EIRP/4π*d2
Note the innocent little "2", and note that d is actually 1018 meters. That's... a mindboggingly huge number, especially after you square it.
Anything "kilo" is too ridiculously close to zero for being measured if you divide it by 1036. Thus, catching any kind of "normal" signal that isn't deliberately directed at us with an extra super strong designed-for-interstellar-hello transmitter is pretty much hopeless.
Certainly, one can more or less arbitrarily scale up the size of the receiving antenna array. Naively, if you don't get enough Watts per square meter out, you just throw in more square meters, and there you go. But the problem is that you also catch more noise. Some of this noise is many orders of magnitude larger than the signal. The chance of actually catching something meaningful is rather dim.
Plus, there's quite a few stars out there. Neglecting everything else, and assuming that we are perfectly able to receive radio from any planet up to 200ly away, there's only about a 1:100,000 chance that we randomly pick the correct star system. Trying them all is possible, of course, but it takes several minutes to rearrange an antenna array, so this task would take anywhere from years to decades.
It might just happen that we only found them now because it took us so long to pick the right star.
Guglielmo Marconi invented the radio in 1894 and it took us less than a decade to realize the need for encryption. At first it was wartime messages only, but it quickly spread to the general populus. Properly tuned cryptography functions are indistinguishable from static, as one would get from a plethora of astonomical bodies or even the vacuum of space. If this species' discovery of radio even remotely resembled our own, we would only have been able to detect their radio chatter for a tiny slice of their history.
This is also one of the main explanations of the Fermi Paradox as to why we haven't found intelligent life yet.
Your alien civilization stopped most of its powerful radio & TV broadcast transmissions as soon as it discovered lasers, fiber optics, and how to send laser pulses over those fibers.
Even when broadcast transmissions were common, the antennae were shaped to keep them hugging the ground (as @Michael Karnerfors mentioned, to do otherwise is wasteful).
We still send transmissions into space, but only ones highly directed at satellites, and then only at the minimum power needed for good reception.
The radio waves are absorbed by our atmosphere. Our choice of radio wave lengths (and vision) is dictated by what can reasonably pass through our atmosphere, if that doesn't line up with what passes through their atmosphere we will not see their ground based transmitters with our ground based receivers.
Schwerin says "radio waves are hard to see," which gave me an idea. These folk have no sense of hearing, and so their writing was pictographic as is their "speech" (sign language). And long before we discovered radio, they had developed to the point that video was preferred over "text."
Although SETI has picked up their transmissions, video of living beings signing inherently doesn't have much pattern, especially when several thousand unrelated signals are mixed together.
Place your planet near a strong astronomical radio source, like a pulsar or other neutron star. The brightness of the pulsar would drown out the relatively week transmissions of a transmitting civilisation unless we were actively looking for their transmissions and knew where to look.
For extra fun, you could combine your pulsar with an active star-forming region, where the hydrogen and other gases that are collapsing into stars are irradiated by the light of earlier stars and release their own radio signals.
Even were you looking in the right direction and for the correct signal, the inverse square law says that the emissions would be detected at extremely weak levels.
A 1W transmitter for example would at 200 light years be received at an energy density of 2 microwatt per square light year. That's not a whole lot for your typical radio telescope that has a size of maybe 400 square meters (for a 20 meter diameter dish, not exactly a tiny one).
And that's the order of magnitude of most radio transmitters in use, even a kilowatt transmitter would only change that signal you receive to 2 milliwatt per square light year.
Remember that a light year is 9.5e15 meters, squared that becomes 9e17 square meters.
Of course that goes out the window if the transmission is highly directional, but the idea holds (as the signal will always spread out, even a laser, parallel by design, will spread out over long distances to where a beam shot from say the moon will have a spot size of several centimeters in diameter when arriving in low earth orbit).
This is an excellent question, and most answers are cool, too. But I'd like to add something.
The thesis that their technology may be far more evolved and thus difficult for us to understand yet is awesome but narrow. I'd say that we do not know how their technology would develop, meaning that they might use different radio transmission than us. As you know, currently all radio is based on Amplitude Modulation, Frequency Modulation and Phase Modulation, meaning that a carrier signal is used to reliably transmit ONLY the amplitude, the frequency or the phase of the transmitted signal as a function of time. That's great for encoding audio in our math system, but their math could look at the universe from a completely different point of view. For all I know, they might not communicate through audio at all (like a giant octopus race mentioned in the Midday World cycle by brothers Strugatsky), thus meaning that they'd transmit only text or other information type. Or they might transmit amplitude and phase simultaneously, for example. The history of our science implies that there is only one way for the science to develop, but (alert: Mass Effect spoiler)
in Mass Effect, for example, Reapers say that their influence on us caused us develop "along the paths they desire" which implies there are ways of development we might be unable to comprehend. So, there's that.
There may be some obstacles for the radiowaves on the border of the Sol system of which we do not know. Imagine that other stars emit radionoise, for example. I am not an expert, but I find it's quite possible. Correct me if I'm wrong :) Or, there could be a black hole that would catch all transmissions into itself.
Given the distance, even 0.01 degree can means a whole different planet altogether.
Plantes rotates around their sun. They are not stationary. Thus you can make it that everytime Earthlings pointed their detection machine in the right direction, there is an object inbetween. For example, another star.
Like others has suggested, since they have the technology years earlier, their encoding could be much more advanced than our own, their radio could use a very high range of frequency that the human's equipments can't capture.
Any number of these make detecting a signal very difficulty, unless one knows the exact code used (or code algorithm), frequency, bandwidth, modulation, and has a high-enough signal to noise ratio. Then there is the actual analog/digital encoding, compression, encryption, symbol set/language, binary/trinary/etc, and more.
SETI says that it is unlikely that a civ would detect us: http://www.seti.org/faq#obs12
Yes - chosen modulation schemes make it difficult to decode the data within the spectrum. But every wireless transmission scheme has a sync method - a way to find the system prior to establishing a connection - which may enable detection - and indicate the PRESENCE of another civilization even if we cannot decode any of the content.
