Balancing Sets of Vectors

For n > 0, d≥ 0, n = d (mod2), let K(n,d) denote the minimal cardinality of a family V of ± 1 vectors of dimension n, such that for any + 1 vector w of dimension n there is a viv such that v·w ≤ d, where v · w is the usual scalar product of v and w. A generalization of a simple construction due to Knuth shows that K(n, d)≤[n/(d + 1)]. A linear algebra proof is given here that this construction is optimal, so that K(n,d) = [n/(d +1)] for all n = d (mod2). This construction and its extensions have applications to communication theory, especially to the construction of signal sets for optical data links.