Gray codes and symmetric chains

We consider the problem of constructing a cyclic listing of all bitstrings of length~$2n+1$ with Hamming weights in the interval $[n+1-\ell,n+\ell]$, where $1\leq \ell\leq n+1$, by flipping a single bit in each step. This is a far-ranging generalization of the well-known middle two levels problem (the case~$\ell=1$). We provide a solution for the case~$\ell=2$ and solve a relaxed version of the problem for general values of~$\ell$, by constructing cycle factors for those instances. Our proof uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets, and we present several new constructions of such decompositions. In particular, we construct four pairwise edge-disjoint symmetric chain decompositions of the $n$-dimensional hypercube for any~$n\geq 12$

On the central levels problem

The central levels problem asserts that the subgraph of the (2m+1)-dimensional hypercube induced by all bitstrings with at least m+1-l many 1s and at most m+l many 1s, i.e., the vertices in the middle 2l levels, has a Hamilton cycle for any m>=1 and 1==1 and 1==2, that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code