Almost Hilarious Poplulation Growth

So In the story I've been working on, there is a five billion strong nation that nobody noticed because they were hiding under Antartica. Now, currently, in the story, the current explanation is that this was an initialy a few families of about a thousand people that balloned into that number in about eighty years.

Now, my question is, can you call BS on this? The society here is primarily dedicated to increasing the population by any means neccessary (for example, it's law there that women capible of pregnancy, must be pregnant as much as possible)... so just assume maximum human fertility being used almost industrialy. Is this possible or do I get to go back to the drawing board?