Discrepancy bounds for infinite-dimensional order two digital sequences over F2F2

We provide explicit constructions of infinite-dimensional digital sequences S=(x0,x1,…)⊂[0,1]NS=(x0,x1,…)⊂[0,1]N, which are constructed over the finite field F2F2, whose projection onto the first s   coordinates x0(s),x1(s),…, for all s⩾1s⩾1, has LqLq discrepancy bounded byLq({x0(s),x1(s),…,xN−1(s)})⩽Cq,sr3/2−1/qN∑v=1rmvs−1 for all N=2m1+2m2+⋯+2mr⩾2N=2m1+2m2+⋯+2mr⩾2 and even integers q   with 2⩽q<∞2⩽q<∞, where the constant Cq,s>0Cq,s>0 is independent of N. In particular, we haveLq({x0(s),x1(s),…,x2m−1(s)})⩽Cq,sm(s−1)/22m for all m,s⩾1m,s⩾1 and 2⩽q<∞2⩽q<∞. Further we give explicit constructions of finite point sets y0,y1,…,yN−1y0,y1,…,yN−1 in [0,1)N[0,1)N for all N⩾2N⩾2 such that their projection on the first s   coordinates y0(s),y1(s),…,yN−1(s) in [0,1)s[0,1)s for all s⩾1s⩾1 satisfiesLq({y0(s),y1(s),…,yN−1(s)})⩽Cq,s(logN)(s−1)/2N for all 2⩽q<∞2⩽q<∞, where Cq,s>0Cq,s>0 is again independent of N. The last two results are best possible by a lower bound of K.F. Roth (1954) [44]. The proofs are based on a generalization of the Niederreiter–Rosenbloom–Tsfasman metric, which itself is a generalization of the Hamming metric.