The mean square discrepancy of randomized nets

One popular family of low dicrepancy sets is the (<italic>t, m, s</italic>)-nets. Recently a randomization of these nets that preserves their net property has been introduced. In this article a formula for the mean square <italic>L</italic><supscrpt>2</supscrpt>-discrepancy of (<italic>0, m, s</italic>)-nets in base <italic>b</italic> is derived. This formula has a computational complexity of only O(s log(<italic>N</italic>) + s<supscrpt>2</supscrpt>) for large <italic>N</italic> or s, where <italic>N = b<supscrpt>m</supscrpt></italic> is the number of points. Moreover, the root mean square <italic>L</italic><supscrpt>2</supscrpt>-discrepancy of (<italic>0, m, s</italic>)-nets is show to be O(<italic>N</italic><supscrpt>-1</supscrpt>[log(N)]<supscrpt>(s-1)/2</supscrpt>) as <italic>N</italic> tends to infinity, the same asymptotic order as the known lower bound for the <italic>L</italic><supscrpt>2</supscrpt>-discrepancy of an arbitrary set.