  We present explicit constructions of centrally symmetric 2-neighborly
d-dimensional polytopes with about 3^{d/2} = (1.73)^d vertices and of centrally
symmetric k-neighborly d-polytopes with about 2^{c_k d} vertices where c_k=3/20
k^2 2^k. Using this result, we construct for a fixed k > 1 and arbitrarily
large d and N, a centrally symmetric d-polytope with N vertices that has at
least (1-k^2 (gamma_k)^d) binom(N, k) faces of dimension k-1, where
gamma_2=1/\sqrt{3} = 0.58 and gamma_k = 2^{-3/{20k^2 2^k}} for k > 2. Another
application is a construction of a set of 3^{d/2 -1}-1 points in R^d every two
of which are strictly antipodal as well as a construction of an n-point set
(for an arbitrarily large n) in R^d with many pairs of strictly antipodal
points. The two latter results significantly improve the previous bounds by
Talata, and Makai and Martini, respectively.
