Obviously, this thread reminds me of the sci-fi novel and movie "Contact" by Carl Sagan. However, I often felt that "Contact" wasn't approaching the idea of contacting an alien civilization(s) effectively. For one the instructions for building an alien machine that can transport one person to the source of the signal wasn't effective at doing what contact is all about and that is introducing oneself! So what would be a better approach? How about sending a message that is simply a "Linux" OS with all kinds of data, even an AI? The first reaction to this is: "But the recipients would need a CPU chip that could decode the instructions of the OS and that conclusion is correct. But, let's not give up on our alien neighbor's sophistication yet. If a civilization can detect the radio signal then they have at least developed electronic technologies to listen for radio. If you were to use a CPU like an x86 or x64 it has over 6000 instructions, but if you use an ARM its instructions set is about 170 or so. Now the challenge is how to teach the aliens the instruction set of an ARM processor! In the movie "Contact" the signal contained mathematical concepts that demonstrated through the use of a simple codification that could be figured out through patterns of zeros and ones, it was binary! So if we focus on mathematical concepts of simple boolean operations and arithmetic we could send information in a signal that conveys binary computations that then can be related to the ARM instruction set. From the instruction set one can devise how to build the hardware, we don't need to teach the aliens how to build computers, we assume that they can. If the aliens have not developed computer technology yet or their computational technology is still in its infancy then the mission has somewhat succeeded in that the aliens know there is another intelligence in the universe! Once their technology has advanced enough they then can build hardware that can load the Linux OS and all the data files sent.
This approach allows for most of the signal sent to be decoded by a machine, and because of that, we can send much richer information like videos, music, virtual worlds, games, videos, models of our understanding of the universe, and an AI. The aliens would have a pretty good understanding of who we are and what we look like and how we think!
There is one more detail I have left out: Realize that the signal being set is graded for different levels of technological sophistication. If we also entangle the radio signal then we and our alien neighbors can connect in real-time, if they have the technological sophistication to do so!