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 correspondence, these ideas are used to derive a dual Plotkin bound for ordered orthogonal arrays. For a (T,M,S)-net in base b, this bound implies TgesM+1-S/1-b^M-Slscr(lscr-1/b-1/b² -middotmiddotmiddot-1/b^lscr), where lscr=1+lfloorM-T/Srfloor. The correspondence ends with an exploration of the implications of this bound relative to known tables and examples