Approximating probability distributions using small sample spaces

We formulate the notion of a "good approximation" to a probability distribu tion over a finite abelian group G. The quality of the approximating distri bution is characterized by a. parameter epsilon which is a bound on the dif ference between corresponding Fourier coefficients of the two distributions . It is also required that the sample space of the approximating distributi on be of size polynomial in log \G\ and 1/epsilon. Such approximations are useful in reducing or eliminating the use of randomness in certain randomiz ed algorithms. We demonstrate the existence of such good approximations to arbitrary distr ibutions. In the case of n random variables distributed uniformly and indep endently over the range {0,...,d-1}, we provide an efficient construction o f a good approximation. The approximation constructed has the property that any linear combination of the random variables (modulo d) has essentially the same behavior under the approximating distribution as it does under the uniform distribution over {0,...,d-1}. Our analysis is based on Well's cha racter sum estimates. We apply this result to the construction of a non-bin ary linear code where the alphabet symbols appear almost uniformly in each non-zero code-word.