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New lower bounds for three- and four-level designs under the cen- tered L2-discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider sev- eral modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered L2-discrepancy. Besides the threshold accepting algorithm, we introduce an algorithm named balance-pursuit heuristic. This algorithm uses some combinatorial properties of inner structures required for a uniform design. Using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works, as well as many new low-discrepancy designs with fairly large scale. Moreover, some of these designs meet the lower bound, i.e., are uniform designs.

Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels