Optimal periodic $L_2$-discrepancy and diaphony bounds for higher order digital sequences

We present an explicit construction of infinite sequences of points $({\mathit{x}}_0,{\mathit{x}}_1,{\mathit{x}}_2,\dots )$ in the $d$-dimensional unit-cube whose periodic $L_2$-discrepancy satisfies $$L_{2,N}^{\mathrm{p}\mathrm{e}\mathrm{r}}(\{{\mathit{x}}_0,{\mathit{x}}_1,\dots ,{\mathit{x}}_{N-1}\})\le C_d{N}^{-1}(\log N{)}^{d\mathrm{/}2}forallN\ge 2,$$ where the factor $C_d>0$ depends only on the dimension $d$. The construction is based on higher order digital sequences as introduced by J. Dick in the year 2008. The result is best possible in the order of magnitude in $N$ according to a Roth-type lower bound shown first by P.D. Proinov. Since the periodic $L_2$-discrepancy is equivalent to P. Zinterhof’s diaphony the result also applies to this alternative quantitative measure for the irregularity of distribution.