Transmitting in the \(n\)-dimensional cube (Q1199408)

It is proved that for any integer \(n\) and for any sequence \(a_ 1,\dots,a_ k\) of \(k=\lceil n/2\rceil\) binary vectors of dimension \(n\) there is a binary vector \(z\) of dimension \(n\) such that the Hamming distance between \(z\) and \(a_ i\) is greater than \(k-i\) for \(i=1,\dots,k\). This result shows that the strategy of the lazy sender on the broadcast problem in the \(n\)-dimension cube with a sender is optimal.