Speaker: Bill Martin (WPI)
Title: The story of (T,M,S)-nets
Abstract: Quasi-Monte Carlo methods are used for numerical integration, simulation, and optimization. In contrast to the true Monte Carlo approach, where points are chosen at random, a quasi-Monte Carlo method employs deterministic point sets which are distributed ``uniformly'' with respect to some pre-specified statistics. These requirements often have combinatorial meaning. Such is the case with (T,M,S)-nets.
This talk focuses on the combinatorics and how I got involved in the subject. Beginning with the definition and small examples, we review the Schmid-Lawrence Theorem which connects (T,M,S)-nets to orthogonal arrays and error-correcting codes. In the 1990s, a puzzle arose as to whether these strange codes have meaningful duals. Doug Stinson and I answered this question and, bringing in the theory of association schemes, introduced the linear programming bound for (T,M,S)-nets, which proved to be quite powerful. The story involves an interesting assortment of researchers in a wonderful mix of geographic locations. So the slides emphasize this social aspect of the tale.