[math-fun] Finding equidistant points among the vertices of a cube

Suppose the metric space Q_n is the vertices of the n-dimensional cube Q_n = {-1, 1}^n with the induced metric from Euclidean space, and the questions is: Question: --------- For which n do the points of Q_n include n+1 equidistant points, the maximum possible? (And in general, what is the size S(n) of the largest equidistant subset of Q_n?) If this is in OEIS, I didn't succeed in finding it. —Dan