  We prove that in all dimensions n at least 3, for every integer N there
exists a distribution of points of cardinality $ N$, for which the associated
discrepancy function D_N satisfies the estimate an estimate, of sharp growth
rate in N, in the exponential Orlicz class exp)L^{2/(n+1)}. This has recently
been proved by M.~Skriganov, using random digit shifts of binary digital nets,
building upon the remarkable examples of W.L.~Chen and M.~Skriganov. Our
approach, developed independently, complements that of Skriganov.
