Simple constructions of almost k-wise independent random variables

The authors present three alternative simple constructions of small probability spaces on n bits for which any k bits are almost independent. The number of bits used to specify a point in the sample space is O(log log n + k + log 1/ε), where ε is the statistical difference between the distribution induced on any k-bit locations and the uniform distribution. This is asymptotically comparable to the construction recently presented by J. Naor and M. Naor (1990). An advantage of the present constructions is their simplicity. Two of the constructions are based on bit sequences that are widely believed to possess randomness properties, and the results can be viewed as an explanation and establishment of these beliefs.