Improved lower bounds on the domination number of hypercubes and binary codes with covering radius one

Abstract

A dominating set on an n-dimensional hypercube is equivalent to a binary covering code of length n and covering radius 1. It is still an open problem to determine the domination number γ(Qn) for n≥10 and n≠2k,2k−1 (k∈N). When n is a multiple of 6, the best known lower bound is γ(Qn)≥2nn, given by Van Wee (1988). In this article, we present a new method using congruence properties due to Laurent Habsieger (1997) and obtain an improved lower bound γ(Qn)≥(n−2)2nn2−2n−2 when n is a multiple of 6.