Before revision:  Unfortunately, the known constructions of weak -nets yields sets that are of size \Omega( ^{-d e^{c d^2} ) } , where c is some constant. We offer two alternative schemes that yield significantly smaller sets, and two related results, as follows: (A)Let be a sample of size ] \tldO(d ^2 ^{-1 , where \tldO hides polylogarithmic terms.
After revision:  Unfortunately, the known constructions of weak -nets yields sets that are of size \Omega( ^{-d e^{c d^2} ) }  be a sample of size ] \tldO(d ^2 ^{-1 , where \tldO hides polylogarithmic terms.