$L_2$ discrepancy of linearly digit scrambled Zaremba point sets

We give an exact formula for the $L_2$ discrepancyof a class of generalized two-dimensional Hammersley point sets in base $b$, namely generalized Zaremba point sets.For the construction of such point sets one needs sequences of permutations of the form $\pi_l(k)=\alpha k +l \pmod{b}$ for $k,l \in {0,\ldots,b-1}$. As a corollary we obtain a condition on these sequences which yields the best possible order of $L_2$ discrepancy of generalized Zaremba point sets in the sense of Roth's lower bound, with very small leading constants.