A Dual Plotkin Bound for (T, M, S) -Nets

The effectiveness of Quasi-Monte Carlo methods for numerical integration has led to the study of (T,M, S)-nets, which are uniformly distributed point sets in the Euclidean unit cube. A recent result, proved independently by Schmid/Mullen and Lawrence, establishes an equivalence between (T,M, S)-nets and ordered orthogonal arrays. In a paper of Martin and Stinson, a linear programming technique is described which gives lower bounds on the size of an ordered orthogonal array and, hence, on the quality parameter T of a (T,M, S)-net. In this paper, we use these ideas to derive a dual Plotkin bound for ordered orthogonal arrays. For a (T,M, S)-net in base b, this bound implies T ≥M + 1− S 1− bM−S` ( `− 1 b − 1 b2 − · · · − 1 b` ) , where ` = 1 + bM−T S c. We end the paper with an exploration of the implications of this bound relative to known tables and examples. 2 MARTIN and VISENTIN