L 2 Discrepancy of Two - Dimensional Digitally Shifted Hammersley Point Sets in Base b

We give an exact formula for the L2 discrepancy of two-dimensional digitally shifted Hammersley point sets in base b. This formula shows that for certain bases b and certain shifts the L2 discrepancy is of best possible order with respect to the general lower bound due to Roth. Hence, for the first time, it is proved that, for a thin, but infinite subsequence of bases b starting with 5, 19, 71, . . ., a single permutation only can achieve this best possible order, unlike previous results of White (1975) who needs b permutations and Faure & Pillichshammer (2008) who need 2 permutations.