Centrally symmetric polytopes with many faces

We present explicit constructions of centrally symmetric polytopes with many faces: (1) we construct a d-dimensional centrally symmetric polytope P with about 3d/4 ≈ (1.316)d vertices such that every pair of non-antipodal vertices of P spans an edge of P , (2) for an integer k ≥ 2, we construct a d-dimensional centrally symmetric polytope P of an arbitrarily high dimension d and with an arbitrarily large number N of vertices such that for some 0 < δk < 1 at least (1 − (δk) d) (N k ) k-subsets of the set of vertices span faces of P , and (3) for an integer k ≥ 2 and α > 0, we construct a centrally symmetric polytope Q with an arbitrarily large number of vertices N and of dimension d = k1+o(1) such that at least ( 1− k−α ) (N k ) k-subsets of the set of vertices span faces of Q.