# How many succinctly generated circuits are there for a given circuit size?

How many circuits are of size $$n$$ are there? In general for a size $$n$$ circuit, I know there are $$O(2^{poly(n)})$$ circuits$$^1$$, but surely this is reduced by the succinctness condition?

$$^1$$ https://cs.stackexchange.com/questions/89488/proving-that-exp-doesnt-have-polynomial-size-circuits

• What exactly do you mean by “succinctly generated circuits”, and how do you measure their size? Note that in general, only $2^{O(n)}$ objects can be described (using whatever fixed encoding scheme) by strings of length $n$ in a finite alphabet. – Emil Jeřábek Feb 8 at 19:02
• Size here meaning the number of gates in the circuit. I appreciate that in general there are only $2^{O(n)}$ circuits for an $n$ bit entry. But does that necessarily tell me anything about the size? Does it straightforwardly imply there are only $log(n)$ circuits with $n$ gates that can be described in any succinct encoding scheme? – user138901 Feb 8 at 20:45
• No, a more likely bound is $n^{O(1)}$ if the encoding produces exponentially larger circuits, but this entirely depends on the encoding scheme, of which you still haven’t specified one iota. – Emil Jeřábek Feb 9 at 7:51