A note on the extreme discrepancy of the Hammersley net in base 2

In this note, we study lower bounds on the extreme discrepancy of the Hammersley net in base 2. The Hammersley netin base 2 can be interpreted as a finite two-dimensional analogue of the well known (one-dimensional) van der Corput sequence in base 2. For the van der Corput sequence it is known that its star discrepancy equals its extreme discrepancy. In this paper, we prove the rather surprising fact that the same does not hold for the Hammersley net, by giving lower bounds on its extreme discrepancy. We furthermore state a few remarks on upper bounds and conclude with a conjecture.