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−`, n+`], where 1 ≤ ` ≤ n+1, by flipping a single bit in each step. This is a far-ranging generalization of the wellknown middle two levels problem (the case ` = 1). We provide a solution for the case ` = 2 and solve a relaxed version of the problem for general values of `, 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 ≥ 12.