Maximal antichains of subsets II: Constructions

This is the second in a sequence of three papers investigating the question for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the previous paper we characterized those $m$ between the maximum size $\binom{n}{\lfloor n/2 \rfloor}$ and $\binom{n}{\lceil n/2\rceil}-\lceil n/2\rceil^2$ that are not sizes of maximal antichains. In this paper we show that all smaller $m$ are sizes of maximal antichains.