Optimal \(\mathcal {L}_{2}\) discrepancy bounds for higher order digital sequences over the finite field \(\mathbb {F}_{2}\) (Q2872668)

For any \(s\in\mathbb{N}\), the authors present an explicit construction of an infinite sequence in the \(s-\)dimensional unit cube such that the \(\mathcal{L}^2\) discrepancy of the first \(N\geq 2\) points is of order \((\log N)^{s/2}/N\) and hence, it is optimal by the Proinov lower bound [\textit{P. D. Proinov}, Serdica 10, 376--383 (1984; Zbl 0598.10054)]. From this result they deduce, for any \(N, s\in\mathbb{N}\) with \(N\geq 2\), an explicit construction of a finite point set of \(N\) elements in \([0,1)^s\) such that the \(\mathcal{L}^2\) discrepancy has convergence rate of order \((\log N)^{(s-1)/2}/N\), which is optimal according to the Roth lower bound [\textit{K. F. Roth}, Mathematika 1, 73--79 (1954; Zbl 0057.28604)]. The latter result was already shown by \textit{W. W. L. Chen} and \textit{M. M. Skriganov} [J. Reine Angew. Math. 545, 67--95 (2002; Zbl 1083.11049)], but with a different method. Indeed, in the paper under review the construction is based on higher order digital sequences and has the advantaged to use only the finite field of order two independently of the dimension \(s\).