I had an old garage door mechanism w/ remote that was turn of the century. There was only one remote (two cars) and was analog. Rather than turfing the whole thing, Linear sold me (through Smarthome.com) a little receiver that activates the manual pushbutton (it's just a receiver + relay) and two remotes. It looks EXACTLY like what you are working with.

From what I can gather, all you need to do is get access to the receiver. On our model, the receiver has a single button, when depressed, puts the unit in "learn" mode then you press the button on the transmitter and the receiver adds the transmitters ID to it's active list. I do not believe they use a rolling code. The receiver accepts up to N transmitter IDs (depends on receiver model), but N is limited, so that's probably why they won't issue another one. They probably only have enough to cover one for every dwelling on the lot (and odds are likely that there is room for a few more).

So if you can get access to the receiver button, buy a new transmitter, all of $20 worst case. Also, inspect the receiver for a model number and look up the manual for it on the web site.

NB - The only thing that doesn't jive with the preceding is the statement that you can open gates at other complexes. So it's possible there is just a single code? Anyway, review the manual for the model you are working with.

GB