Optimality of the reverse-priority rule for all hypercube dimensions

Establish that, for every integer n ≥ 1, among perfect matchings on the Boolean hypercube {0,1}^n that are equivariant under the Klein four-group generated by bitwise complement comp and bit reversal rev and that restrict each pair to either {h, comp(h)} or {h, rev(h)}, the matching defined by the reverse-priority rule—pair each h with rev(h) if h ≠ rev(h), and otherwise pair h with comp(h)—minimizes the total Hamming cost.

Background

The paper studies K4-equivariant perfect matchings on the 6-dimensional Boolean hypercube {0,1}^6, focusing on matchings that pair each element using either bitwise complement (comp) or bit reversal (rev). It proves that for n = 6 there is a unique cost-minimizing matching under this restriction, given by a simple reverse-priority rule: pair with reversal unless the element is a palindrome (rev(h) = h), in which case pair with complement.

The authors further show that allowing comp ∘ rev pairings can reduce the total cost below the reverse-priority rule for n = 6, but such pairings lack a uniform rule and require case-by-case analysis. Motivated by the n = 6 results, they pose a conjecture about the optimality of the reverse-priority rule for all dimensions n under the same comp/rev pairing restriction.