Define the partial order on a set $X$ of all binary strings of length n to be $xRy$ if and only if $x=y$ or $x$ has an odd number of ones and $x$ and $y$ are adjacent vertices on the hypercube $Q_n$, ie differ in exactly one place. I'm able to show this is a partial order easily enough, but I'm struggling to find a maximal antichain $A$ on this poset, and I'm also struggling to find a minimum chain covering with $|A|$ chains.
I've written out the partial order for $n=3$ but still am not getting anywhere, even with finding an antichain.