We show that the $\mathcal{ L}_2$ discrepancy of the explicitly constructed infinite sequences of points $(\boldsymbol{ x}_0,\boldsymbol{ x}_1, \boldsymbol{ x}_2,...)$ in $[0,1)^s$ over $\mathbb{ F}_2$ introduced in [J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order. SIAM J. Numer. Anal., { \bf 46}, 1519--1553, 2008] satisfy $$\mathcal{ L}_{ 2,N}(\{ \boldsymbol{ x}_0,\boldsymbol{ x}_1,..., \boldsymbol{ x}_{ N-1}\}) \le C_s N^{ -1} (\log N)^{ s/2} \quad { for all} N \ge 2,$$ and $$\mathcal{ L}_{ 2,2^m}(\{ \boldsymbol{ x}_0,\boldsymbol{ x}_1,..., \boldsymbol{ x}_{ 2^m-1}\}) \le C_s 2^{ -m} m^{ (s-1)/2} \quad { for all} m \ge 1,$$ where $C_s > 0$ is a constant independent of $N$ and $m$. These results are best possible by lower bounds in [P.D. Proinov, On the $L^2$ discrepancy of some infinite sequences. Serdica, { \bf 11}, 3--12, 1985] and [K. F. Roth, On irregularities of distribution. Mathematika, { \bf 1}, 73--79, 1954]. Further, for every $N \ge 2$ we explicitly construct finite point sets $\{ \boldsymbol{ y}_0,..., \boldsymbol{ y}_{ N-1}\}$ in $[0,1)^s$ such that $$\mathcal{ L}_{ 2,N}(\{ \boldsymbol{ y}_0,\boldsymbol{ y}_1,..., \boldsymbol{ y}_{ N-1}\}) \le C_s N^{ -1} (\log N)^{ (s-1)/2}.$$ Another solution for finite point sets by a different construction was previously shown in [W. W. L. Chen and M. M. Skriganov, Explicit constructions in the classical mean squares problem in irregularity of point distribution. J. Reine Angew. Math., { \bf 545}, 67--95, 2002].
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