Let n=2k−1, where k≥6 is an integer. Let T be the set of all n-tuples <spanclass=′latex−bold′>x</span>=(x1,x2,…,xn), where, for i=1,2,…,n,xi is 0 or 1. For <spanclass=′latex−bold′>x</span>=(x1,x2,…,xn) and <spanclass=′latex−bold′>y</span>=(y1,y2,…,yn) in T, let d(<spanclass=′latex−bold′>x</span>,<spanclass=′latex−bold′>y</span>) denote the number of integers j with 1≤j≤n such that xj=xy. (In particular, d(<spanclass=′latex−bold′>x</span>,<spanclass=′latex−bold′>x</span>)=0).Suppose that there exists a subset S of T with 2k elements which has the following property: given any element <spanclass=′latex−bold′>x</span> in T, there is a unique <spanclass=′latex−bold′>y</span> in S with d(<spanclass=′latex−bold′>x</span>,<spanclass=′latex−bold′>y</span>)≤3.Prove that n=23.