On the existence of symmetric chain decompositions in a quotient of the Boolean lattice

We highlight a question about binary necklaces, i.e., equivalence classes of binary strings under rotation. Is there a way to choose representatives of the nn-bit necklaces so that the subposet of the Boolean lattice induced by those representatives has a symmetric chain decomposition? Alternatively, is the quotient of the Boolean lattice BnBn, under the action of the cyclic group ZnZn, a symmetric chain order? The answer is known to be yes for all prime nn and for composite n≤18n≤18, but otherwise the question appears to be open. In this note we describe how it suffices to focus on subposets induced by necklaces with periodic block codes, substantially reducing the size of the problem. We mention a motivating application: determining whether minimum-region rotationally symmetric independent families of nn curves exist for all nn.