L2 discrepancy of symmetrized generalized Hammersley point sets in base b

Abstract

Two popular and often applied methods to obtain two-dimensional point sets with the optimal order of Lp discrepancy are digit scrambling and symmetrization. In this paper we combine these two techniques and symmetrize b-adic Hammersley point sets scrambled with arbitrary permutations. It is already known that these modifications indeed assure that the Lp discrepancy is of optimal order O(log⁡N/N) for p∈[1,∞) in contrast to the classical Hammersley point set. We prove an exact formula for the L2 discrepancy of these point sets for special permutations. We also present the permutations which lead to the lowest L2 discrepancy for every base b∈{2,…,27} by employing computer search algorithms.