We will present a Kashin type result for embedding lpn into l1N for 1 < p < 2 and N arbitrarily close to n. We will show that it is possible to define random embedings such that the conclusion holds with overwhelming probability. The result can also be extended to embedding from lpn to lrN with 0 < r = 1. One of the main tool that we developp is a new type of multi-dimensional Esseen inequality for studying small ball probabilities.