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Efficient Uniform k-out-of-n Generators

In many distributed network problems, one has to randomly choose a subset of servers in order to execute a task. This can be achieved by using a so called k-out-of-n generator: a generator which randomly chooses k elements among n elements. We introduce new constructions of uniform k-out-of-n generators from a binary generator taken as a primary source of alea. These constructions make use of special codes containing combinatorial objects called Steiner systems. Any Steiner system leads to a generator having a maximal entropy rate. As an example, we analyse in detail the special case k=5 and n=24 and we study the convergence of a random walk on the Mathieu group to the uniform distribution. We show that the speed of convergence is excellent, and as far as we know, better than any known methods.