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In previous work (Kuo, Nuyens, Wilkes, 2023), we showed that a lattice rulewith a pre-determined generating vector but random number of points can achievethe near optimal convergence of $O(n^{-\alpha-1/2+\epsilon})$, $\epsilon > 0$,for the worst case expected error, commonly referred to as the randomisederror, for numerical integration of high-dimensional functions in the Korobovspace with smoothness $\alpha > 1/2$. Compared to the optimal deterministicrate of $O(n^{-\alpha+\epsilon})$, $\epsilon > 0$, such a randomised algorithmis capable of an extra half in the rate of convergence. In this paper, we showthat a pre-determined generating vector also exists in the case of $0 < \alpha\le 1/2$. Also here we obtain the near optimal convergence of$O(n^{-\alpha-1/2+\epsilon})$, $\epsilon > 0$; or in more detail, we obtain$O(\sqrt{r} \, n^{-\alpha-1/2+1/(2r)+\epsilon'})$ which holds for any choicesof $\epsilon' > 0$ and $r \in \mathbb{N}$ with $r > 1/(2\alpha)$.

A randomised lattice rule algorithm with pre-determined generating vector and random number of points for Korobov spaces with $0 < \alpha \le 1/2$