On explicit formulas for bandwidth and antibandwidth of hypercubes

The Hales numbered n-dimensional hypercube exhibits interesting recursive structures in n. These structures lead to a very simple proof of the well-known bandwidth formula for hypercubes proposed by Harper, whose proof was thought to be surprisingly difficult. Harper also proposed an optimal numbering for a related problem called the antibandwidth of hypercubes. In a recent publication, Raspaud et al. approximated the hypercube antibandwidth up to the third-order term. In this paper, we find the exact value in light of the above recursive structures.