For each non-negative integer $n$, what antichain(s) in $\{0,1\}^n$ with the pointwise partial order:

$\;\;$ 1. $\;$ have the most elements

$\;\;$ 2. $\;$ minimize the maximum of its elements' sum of coordinates, among those satisfying (1)

?

The obvious candidate is the set of elements whose sum of coordinates is $\: \Big\lfloor \frac{n}2 \Big\rfloor \:$.

Where $S$ is such a set and $\: m = \big\lfloor \log_2(|S|) \big\rfloor \:$, $\;$ I would also want an

efficiently computable injection $\: f : \{0,1\}^m \to S \:$ whose inverse is efficiently computable,

although I imagine that part would be straight-forward.