Family Register, Census
noun
Enroll your door... and by door, I mean enroll your family, because each family has one door to their house. Enrolling your family means you are enrolling them into the family register. This is actually an interesting topic. Look up "koseki" in Google and you're sure to find many fascinating histories.
こせき
The reading for 戸 is an exception. Instead of と it's こ. Just think about how the person who runs this census doesn't care about number of toes. They care about number of 子 (こ).
私は戸籍謄本の写しが必要です。
I need an official copy of my family register.