 at least $0^{\Omega(n)}$.  By comparing the complexities of the circuit and the polynomial as given by Corollary \[cor:circuit-complexity-bound\] and Lemma \[lem:manyshiftsNW\], the top fan-in of the circuit must be at least  $$~\label{eqn:top-fan-in-0} \frac{0.0n^{r}{N\choose m}{\ell-0 \choose m-0}}{\text{poly}(nrs){n+r \choose r}{N \choose m+rs}{\ell+n-r \choose m+rs}}$$  This bound holds for any choice of positive integers $\ell, m, r$, a constant $\beta$ such that $s = \beta\log n$ which satisfy the constraints in the hypothesis of Corollary \[cor:circuit-complexity-bound\] and Lemma \[lem:manyshiftsNW\]. In other words, we want these parameters to satisfy  -   $m+rs \leq \frac{N}{0}$  -   $m+rs \leq \frac{\ell}{0}$  -   $