There is also a papyrus from about 80 AD in Egypt telling of a census there, in which everyone was ordered to return to their home area for counting.
We date the death of Herod at 4 BC, because Josephus says he died shortly after an eclipse of the moon. But there were a couple of partial eclipses after that, so that it is possible Herod lived past 4 BC and died later.
There is good evidence of a worldwide tax imposed by Augustus around 3 BC, in which people also pledged allegiance to him. That may very well be the census referred to by Luke (and not an early one in 7 BC, which
actually was only of Roman citizens).