Gray codes and symmetric chains
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We consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1-ℓ,n+ℓ], where 1≤ℓ≤ 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 ℓ=1). We provide a solution for the case ℓ=2 and solve a relaxed version of the problem for general values of ℓ, 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.
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